:: CATALAN2 semantic presentation begin definition let p, q be XFinSequence of ; :: original: ^ redefine funcp ^ q -> XFinSequence of ; coherence p ^ q is XFinSequence of ; end; theorem Th1: :: CATALAN2:1 for D being set for n being Nat for pd being XFinSequence of ex qd being XFinSequence of st pd = (pd | n) ^ qd proof let D be set ; ::_thesis: for n being Nat for pd being XFinSequence of ex qd being XFinSequence of st pd = (pd | n) ^ qd let n be Nat; ::_thesis: for pd being XFinSequence of ex qd being XFinSequence of st pd = (pd | n) ^ qd let pd be XFinSequence of ; ::_thesis: ex qd being XFinSequence of st pd = (pd | n) ^ qd consider q9 being XFinSequence such that A1: pd = (pd | n) ^ q9 by AFINSQ_1:60; rng q9 c= rng pd by A1, AFINSQ_1:25; then rng q9 c= D by XBOOLE_1:1; then q9 is XFinSequence of by RELAT_1:def_19; hence ex qd being XFinSequence of st pd = (pd | n) ^ qd by A1; ::_thesis: verum end; definition let p be XFinSequence of ; attrp is dominated_by_0 means :Def1: :: CATALAN2:def 1 ( rng p c= {0,1} & ( for k being Nat st k <= dom p holds 2 * (Sum (p | k)) <= k ) ); end; :: deftheorem Def1 defines dominated_by_0 CATALAN2:def_1_:_ for p being XFinSequence of holds ( p is dominated_by_0 iff ( rng p c= {0,1} & ( for k being Nat st k <= dom p holds 2 * (Sum (p | k)) <= k ) ) ); theorem Th2: :: CATALAN2:2 for k being Nat for p being XFinSequence of st p is dominated_by_0 holds 2 * (Sum (p | k)) <= k proof let k be Nat; ::_thesis: for p being XFinSequence of st p is dominated_by_0 holds 2 * (Sum (p | k)) <= k let p be XFinSequence of ; ::_thesis: ( p is dominated_by_0 implies 2 * (Sum (p | k)) <= k ) assume A1: p is dominated_by_0 ; ::_thesis: 2 * (Sum (p | k)) <= k now__::_thesis:_2_*_(Sum_(p_|_k))_<=_k percases ( k <= dom p or k > dom p ) ; suppose k <= dom p ; ::_thesis: 2 * (Sum (p | k)) <= k hence 2 * (Sum (p | k)) <= k by A1, Def1; ::_thesis: verum end; supposeA2: k > dom p ; ::_thesis: 2 * (Sum (p | k)) <= k then dom p c= k by NAT_1:39; then A3: p | k = p by RELAT_1:68; ( 2 * (Sum (p | (len p))) <= dom p & p | (len p) = p ) by A1, Def1, RELAT_1:68; hence 2 * (Sum (p | k)) <= k by A2, A3, XXREAL_0:2; ::_thesis: verum end; end; end; hence 2 * (Sum (p | k)) <= k ; ::_thesis: verum end; theorem Th3: :: CATALAN2:3 for p being XFinSequence of st p is dominated_by_0 holds p . 0 = 0 proof let p be XFinSequence of ; ::_thesis: ( p is dominated_by_0 implies p . 0 = 0 ) assume A1: p is dominated_by_0 ; ::_thesis: p . 0 = 0 now__::_thesis:_p_._0_=_0 percases ( not 0 in dom p or 0 in dom p ) ; suppose not 0 in dom p ; ::_thesis: p . 0 = 0 hence p . 0 = 0 by FUNCT_1:def_2; ::_thesis: verum end; suppose 0 in dom p ; ::_thesis: p . 0 = 0 then len p >= 1 by NAT_1:14; then A2: 1 c= len p by NAT_1:39; 0 in 1 by NAT_1:44; then 0 in (dom p) /\ 1 by A2, XBOOLE_0:def_4; then A3: (p | 1) . 0 = p . 0 by FUNCT_1:48; A4: Sum <%(p . 0)%> = addnat "**" <%(p . 0)%> by AFINSQ_2:51 .= p . 0 by AFINSQ_2:37 ; len (p | 1) = 1 by A2, RELAT_1:62; then p | 1 = <%(p . 0)%> by A3, AFINSQ_1:34; then 2 * (p . 0) <= 1 + 0 by A1, A4, Th2; then ( 2 * (p . 0) = 1 or 2 * (p . 0) = 0 ) by NAT_1:9; hence p . 0 = 0 by NAT_1:15; ::_thesis: verum end; end; end; hence p . 0 = 0 ; ::_thesis: verum end; registration let x be set ; let k be Nat; clusterx --> k -> NAT -valued ; coherence x --> k is NAT -valued proof k in NAT by ORDINAL1:def_12; then ( rng (x --> k) c= {k} & {k} c= NAT ) by ZFMISC_1:31; then rng (x --> k) c= NAT by XBOOLE_1:1; hence x --> k is NAT -valued by RELAT_1:def_19; ::_thesis: verum end; end; Lm1: for n, m, k being Nat st n <= m holds (m --> k) | n = n --> k proof let n, m, k be Nat; ::_thesis: ( n <= m implies (m --> k) | n = n --> k ) assume n <= m ; ::_thesis: (m --> k) | n = n --> k then n c= m by NAT_1:39; then n /\ m = n by XBOOLE_1:28; hence (m --> k) | n = n --> k by FUNCOP_1:12; ::_thesis: verum end; Lm2: for k being Nat holds k --> 0 is dominated_by_0 proof let k be Nat; ::_thesis: k --> 0 is dominated_by_0 A1: dom (k --> 0) = k by FUNCOP_1:13; ( rng (k --> 0) c= {0} & {0} c= {0,1} ) by FUNCOP_1:13, ZFMISC_1:7; hence rng (k --> 0) c= {0,1} by XBOOLE_1:1; :: according to CATALAN2:def_1 ::_thesis: for k being Nat st k <= dom (k --> 0) holds 2 * (Sum ((k --> 0) | k)) <= k let n be Nat; ::_thesis: ( n <= dom (k --> 0) implies 2 * (Sum ((k --> 0) | n)) <= n ) assume n <= dom (k --> 0) ; ::_thesis: 2 * (Sum ((k --> 0) | n)) <= n then A2: (k --> 0) | n = n --> 0 by A1, Lm1; Sum (n --> 0) = 0 * n by AFINSQ_2:58; hence 2 * (Sum ((k --> 0) | n)) <= n by A2; ::_thesis: verum end; registration cluster empty T-Sequence-like Relation-like NAT -defined NAT -valued Function-like V35() V36() V37() V38() finite nonnegative-yielding V213() dominated_by_0 for set ; existence ex b1 being XFinSequence of st ( b1 is empty & b1 is dominated_by_0 ) proof 0 --> 0 is dominated_by_0 by Lm2; hence ex b1 being XFinSequence of st ( b1 is empty & b1 is dominated_by_0 ) ; ::_thesis: verum end; cluster non empty T-Sequence-like Relation-like NAT -defined NAT -valued Function-like V35() V36() V37() V38() finite nonnegative-yielding V213() dominated_by_0 for set ; existence ex b1 being XFinSequence of st ( not b1 is empty & b1 is dominated_by_0 ) proof 1 --> 0 is dominated_by_0 by Lm2; hence ex b1 being XFinSequence of st ( not b1 is empty & b1 is dominated_by_0 ) ; ::_thesis: verum end; end; theorem :: CATALAN2:4 for n being Nat holds n --> 0 is dominated_by_0 by Lm2; theorem Th5: :: CATALAN2:5 for n, m being Nat st n >= m holds (n --> 0) ^ (m --> 1) is dominated_by_0 proof let n, m be Nat; ::_thesis: ( n >= m implies (n --> 0) ^ (m --> 1) is dominated_by_0 ) assume A1: n >= m ; ::_thesis: (n --> 0) ^ (m --> 1) is dominated_by_0 set p = (n --> 0) ^ (m --> 1); ( rng (m --> 1) c= {1} & {1} c= {0,1} ) by FUNCOP_1:13, ZFMISC_1:7; then A2: rng (m --> 1) c= {0,1} by XBOOLE_1:1; ( rng (n --> 0) c= {0} & {0} c= {0,1} ) by FUNCOP_1:13, ZFMISC_1:7; then rng (n --> 0) c= {0,1} by XBOOLE_1:1; then (rng (n --> 0)) \/ (rng (m --> 1)) c= {0,1} by A2, XBOOLE_1:8; hence rng ((n --> 0) ^ (m --> 1)) c= {0,1} by AFINSQ_1:26; :: according to CATALAN2:def_1 ::_thesis: for k being Nat st k <= dom ((n --> 0) ^ (m --> 1)) holds 2 * (Sum (((n --> 0) ^ (m --> 1)) | k)) <= k let k be Nat; ::_thesis: ( k <= dom ((n --> 0) ^ (m --> 1)) implies 2 * (Sum (((n --> 0) ^ (m --> 1)) | k)) <= k ) assume A3: k <= dom ((n --> 0) ^ (m --> 1)) ; ::_thesis: 2 * (Sum (((n --> 0) ^ (m --> 1)) | k)) <= k now__::_thesis:_2_*_(Sum_(((n_-->_0)_^_(m_-->_1))_|_k))_<=_k percases ( k <= dom (n --> 0) or k > dom (n --> 0) ) ; supposeA4: k <= dom (n --> 0) ; ::_thesis: 2 * (Sum (((n --> 0) ^ (m --> 1)) | k)) <= k dom (n --> 0) = n by FUNCOP_1:13; then A5: (n --> 0) | k = k --> 0 by A4, Lm1; A6: Sum (k --> 0) = 0 * k by AFINSQ_2:58; ((n --> 0) ^ (m --> 1)) | k = (n --> 0) | k by A4, AFINSQ_1:58; hence 2 * (Sum (((n --> 0) ^ (m --> 1)) | k)) <= k by A5, A6; ::_thesis: verum end; suppose k > dom (n --> 0) ; ::_thesis: 2 * (Sum (((n --> 0) ^ (m --> 1)) | k)) <= k then reconsider kd = k - (dom (n --> 0)) as Element of NAT by NAT_1:21; A7: dom (n --> 0) = n by FUNCOP_1:13; dom ((n --> 0) ^ (m --> 1)) = len ((n --> 0) ^ (m --> 1)) ; then k <= (len (n --> 0)) + (len (m --> 1)) by A3, AFINSQ_1:17; then k - (len (n --> 0)) <= ((len (m --> 1)) + (len (n --> 0))) - (len (n --> 0)) by XREAL_1:9; then kd <= m by FUNCOP_1:13; then A8: (m --> 1) | kd = kd --> 1 by Lm1; reconsider m1 = m --> 1 as XFinSequence of ; k = kd + (dom (n --> 0)) ; then ((n --> 0) ^ (m --> 1)) | k = (n --> 0) ^ (m1 | kd) by AFINSQ_1:59; then A9: Sum (((n --> 0) ^ (m --> 1)) | k) = (Sum (n --> 0)) + (Sum (kd --> 1)) by A8, AFINSQ_2:55; ( dom ((n --> 0) ^ (m --> 1)) = (len (n --> 0)) + (len (m --> 1)) & dom (m --> 1) = m ) by AFINSQ_1:def_3, FUNCOP_1:13; then k - n <= (m + n) - n by A3, A7, XREAL_1:9; then k - n <= n by A1, XXREAL_0:2; then A10: (k - n) + (k - n) <= n + (k - n) by XREAL_1:6; ( Sum (n --> 0) = n * 0 & Sum (kd --> 1) = kd * 1 ) by AFINSQ_2:58; hence 2 * (Sum (((n --> 0) ^ (m --> 1)) | k)) <= k by A9, A7, A10; ::_thesis: verum end; end; end; hence 2 * (Sum (((n --> 0) ^ (m --> 1)) | k)) <= k ; ::_thesis: verum end; theorem Th6: :: CATALAN2:6 for n being Nat for p being XFinSequence of st p is dominated_by_0 holds p | n is dominated_by_0 proof let n be Nat; ::_thesis: for p being XFinSequence of st p is dominated_by_0 holds p | n is dominated_by_0 let p be XFinSequence of ; ::_thesis: ( p is dominated_by_0 implies p | n is dominated_by_0 ) assume A1: p is dominated_by_0 ; ::_thesis: p | n is dominated_by_0 A2: for k being Nat st k <= dom (p | n) holds 2 * (Sum ((p | n) | k)) <= k proof let k be Nat; ::_thesis: ( k <= dom (p | n) implies 2 * (Sum ((p | n) | k)) <= k ) assume k <= dom (p | n) ; ::_thesis: 2 * (Sum ((p | n) | k)) <= k then A3: k c= dom (p | n) by NAT_1:39; dom (p | n) = (dom p) /\ n by RELAT_1:61; then (p | n) | k = p | k by A3, RELAT_1:74, XBOOLE_1:18; hence 2 * (Sum ((p | n) | k)) <= k by A1, Th2; ::_thesis: verum end; ( rng (p | n) c= rng p & rng p c= {0,1} ) by A1, Def1, RELAT_1:70; then rng (p | n) c= {0,1} by XBOOLE_1:1; hence p | n is dominated_by_0 by A2, Def1; ::_thesis: verum end; theorem Th7: :: CATALAN2:7 for p, q being XFinSequence of st p is dominated_by_0 & q is dominated_by_0 holds p ^ q is dominated_by_0 proof let p, q be XFinSequence of ; ::_thesis: ( p is dominated_by_0 & q is dominated_by_0 implies p ^ q is dominated_by_0 ) assume that A1: p is dominated_by_0 and A2: q is dominated_by_0 ; ::_thesis: p ^ q is dominated_by_0 ( rng p c= {0,1} & rng q c= {0,1} ) by A1, A2, Def1; then (rng p) \/ (rng q) c= {0,1} by XBOOLE_1:8; hence rng (p ^ q) c= {0,1} by AFINSQ_1:26; :: according to CATALAN2:def_1 ::_thesis: for k being Nat st k <= dom (p ^ q) holds 2 * (Sum ((p ^ q) | k)) <= k let k be Nat; ::_thesis: ( k <= dom (p ^ q) implies 2 * (Sum ((p ^ q) | k)) <= k ) assume k <= dom (p ^ q) ; ::_thesis: 2 * (Sum ((p ^ q) | k)) <= k now__::_thesis:_2_*_(Sum_((p_^_q)_|_k))_<=_k percases ( k <= dom p or k > dom p ) ; supposeA3: k <= dom p ; ::_thesis: 2 * (Sum ((p ^ q) | k)) <= k then (p ^ q) | k = p | k by AFINSQ_1:58; hence 2 * (Sum ((p ^ q) | k)) <= k by A1, A3, Def1; ::_thesis: verum end; suppose k > dom p ; ::_thesis: 2 * (Sum ((p ^ q) | k)) <= k then reconsider kd = k - (dom p) as Element of NAT by NAT_1:21; A4: p | (dom p) = p ; k = kd + (dom p) ; then (p ^ q) | k = p ^ (q | kd) by AFINSQ_1:59; then A5: Sum ((p ^ q) | k) = (Sum p) + (Sum (q | kd)) by AFINSQ_2:55; ( 2 * (Sum (p | (len p))) <= len p & 2 * (Sum (q | kd)) <= kd ) by A1, A2, Th2; then (2 * (Sum p)) + (2 * (Sum (q | kd))) <= (dom p) + kd by A4, XREAL_1:7; hence 2 * (Sum ((p ^ q) | k)) <= k by A5; ::_thesis: verum end; end; end; hence 2 * (Sum ((p ^ q) | k)) <= k ; ::_thesis: verum end; theorem Th8: :: CATALAN2:8 for n being Nat for p being XFinSequence of st p is dominated_by_0 holds 2 * (Sum (p | ((2 * n) + 1))) < (2 * n) + 1 proof let n be Nat; ::_thesis: for p being XFinSequence of st p is dominated_by_0 holds 2 * (Sum (p | ((2 * n) + 1))) < (2 * n) + 1 let p be XFinSequence of ; ::_thesis: ( p is dominated_by_0 implies 2 * (Sum (p | ((2 * n) + 1))) < (2 * n) + 1 ) assume p is dominated_by_0 ; ::_thesis: 2 * (Sum (p | ((2 * n) + 1))) < (2 * n) + 1 then A1: 2 * (Sum (p | ((2 * n) + 1))) <= (2 * n) + 1 by Th2; assume 2 * (Sum (p | ((2 * n) + 1))) >= (2 * n) + 1 ; ::_thesis: contradiction then 2 * (Sum (p | ((2 * n) + 1))) = (2 * n) + 1 by A1, XXREAL_0:1; then 2 * ((Sum (p | ((2 * n) + 1))) - n) = 1 ; hence contradiction by INT_1:9; ::_thesis: verum end; theorem Th9: :: CATALAN2:9 for n being Nat for p being XFinSequence of st p is dominated_by_0 & n <= (len p) - (2 * (Sum p)) holds p ^ (n --> 1) is dominated_by_0 proof let n be Nat; ::_thesis: for p being XFinSequence of st p is dominated_by_0 & n <= (len p) - (2 * (Sum p)) holds p ^ (n --> 1) is dominated_by_0 let p be XFinSequence of ; ::_thesis: ( p is dominated_by_0 & n <= (len p) - (2 * (Sum p)) implies p ^ (n --> 1) is dominated_by_0 ) set q = n --> 1; assume that A1: p is dominated_by_0 and A2: n <= (len p) - (2 * (Sum p)) ; ::_thesis: p ^ (n --> 1) is dominated_by_0 ( rng (n --> 1) c= {1} & {1} c= {0,1} ) by FUNCOP_1:13, ZFMISC_1:7; then A3: rng (n --> 1) c= {0,1} by XBOOLE_1:1; rng p c= {0,1} by A1, Def1; then (rng p) \/ (rng (n --> 1)) c= {0,1} by A3, XBOOLE_1:8; hence rng (p ^ (n --> 1)) c= {0,1} by AFINSQ_1:26; :: according to CATALAN2:def_1 ::_thesis: for k being Nat st k <= dom (p ^ (n --> 1)) holds 2 * (Sum ((p ^ (n --> 1)) | k)) <= k let m be Nat; ::_thesis: ( m <= dom (p ^ (n --> 1)) implies 2 * (Sum ((p ^ (n --> 1)) | m)) <= m ) assume A4: m <= dom (p ^ (n --> 1)) ; ::_thesis: 2 * (Sum ((p ^ (n --> 1)) | m)) <= m now__::_thesis:_2_*_(Sum_((p_^_(n_-->_1))_|_m))_<=_m percases ( m <= dom p or m > dom p ) ; suppose m <= dom p ; ::_thesis: 2 * (Sum ((p ^ (n --> 1)) | m)) <= m then (p ^ (n --> 1)) | m = p | m by AFINSQ_1:58; hence 2 * (Sum ((p ^ (n --> 1)) | m)) <= m by A1, Th2; ::_thesis: verum end; suppose m > dom p ; ::_thesis: 2 * (Sum ((p ^ (n --> 1)) | m)) <= m then reconsider md = m - (dom p) as Element of NAT by NAT_1:21; A5: m = (dom p) + md ; Sum (md --> 1) = md * 1 by AFINSQ_2:58; then A6: Sum (p ^ (md --> 1)) = (Sum p) + md by AFINSQ_2:55; ( dom (n --> 1) = n & len (n --> 1) = dom (n --> 1) ) by FUNCOP_1:13; then dom (p ^ (n --> 1)) = (len p) + n by AFINSQ_1:def_3; then md + (dom p) <= n + (dom p) by A4; then A7: md <= n by XREAL_1:6; then (n --> 1) | md = md --> 1 by Lm1; then (p ^ (n --> 1)) | m = p ^ (md --> 1) by A5, AFINSQ_1:59; then 2 * (Sum ((p ^ (n --> 1)) | m)) = (((2 * (Sum p)) + m) - (dom p)) + md by A6; then A8: 2 * (Sum ((p ^ (n --> 1)) | m)) <= (((2 * (Sum p)) + m) - (dom p)) + n by A7, XREAL_1:6; n - n <= ((len p) - (2 * (Sum p))) - n by A2, XREAL_1:9; then m - (((len p) - (2 * (Sum p))) - n) <= m - 0 by XREAL_1:10; hence 2 * (Sum ((p ^ (n --> 1)) | m)) <= m by A8, XXREAL_0:2; ::_thesis: verum end; end; end; hence 2 * (Sum ((p ^ (n --> 1)) | m)) <= m ; ::_thesis: verum end; theorem Th10: :: CATALAN2:10 for n, k being Nat for p being XFinSequence of st p is dominated_by_0 & n <= (k + (len p)) - (2 * (Sum p)) holds ((k --> 0) ^ p) ^ (n --> 1) is dominated_by_0 proof let n, k be Nat; ::_thesis: for p being XFinSequence of st p is dominated_by_0 & n <= (k + (len p)) - (2 * (Sum p)) holds ((k --> 0) ^ p) ^ (n --> 1) is dominated_by_0 let p be XFinSequence of ; ::_thesis: ( p is dominated_by_0 & n <= (k + (len p)) - (2 * (Sum p)) implies ((k --> 0) ^ p) ^ (n --> 1) is dominated_by_0 ) assume that A1: p is dominated_by_0 and A2: n <= (k + (len p)) - (2 * (Sum p)) ; ::_thesis: ((k --> 0) ^ p) ^ (n --> 1) is dominated_by_0 set q = k --> 0; ( dom (k --> 0) = k & len (k --> 0) = dom (k --> 0) ) by FUNCOP_1:13; then A3: len ((k --> 0) ^ p) = k + (len p) by AFINSQ_1:17; Sum (k --> 0) = k * 0 by AFINSQ_2:58; then A4: Sum ((k --> 0) ^ p) = 0 + (Sum p) by AFINSQ_2:55; k --> 0 is dominated_by_0 by Lm2; then (k --> 0) ^ p is dominated_by_0 by A1, Th7; hence ((k --> 0) ^ p) ^ (n --> 1) is dominated_by_0 by A2, A3, A4, Th9; ::_thesis: verum end; theorem Th11: :: CATALAN2:11 for k being Nat for p being XFinSequence of st p is dominated_by_0 & 2 * (Sum (p | k)) = k holds ( k <= len p & len (p | k) = k ) proof let k be Nat; ::_thesis: for p being XFinSequence of st p is dominated_by_0 & 2 * (Sum (p | k)) = k holds ( k <= len p & len (p | k) = k ) let p be XFinSequence of ; ::_thesis: ( p is dominated_by_0 & 2 * (Sum (p | k)) = k implies ( k <= len p & len (p | k) = k ) ) assume A1: ( p is dominated_by_0 & 2 * (Sum (p | k)) = k ) ; ::_thesis: ( k <= len p & len (p | k) = k ) A2: k <= len p proof A3: p | (len p) = p by RELAT_1:68; assume A4: k > len p ; ::_thesis: contradiction then len p c= k by NAT_1:39; then p | k = p by RELAT_1:68; hence contradiction by A1, A4, A3, Th2; ::_thesis: verum end; then k c= len p by NAT_1:39; then (dom p) /\ k = k by XBOOLE_1:28; hence ( k <= len p & len (p | k) = k ) by A2, RELAT_1:61; ::_thesis: verum end; theorem Th12: :: CATALAN2:12 for k being Nat for p, q being XFinSequence of st p is dominated_by_0 & 2 * (Sum (p | k)) = k & p = (p | k) ^ q holds q is dominated_by_0 proof let k be Nat; ::_thesis: for p, q being XFinSequence of st p is dominated_by_0 & 2 * (Sum (p | k)) = k & p = (p | k) ^ q holds q is dominated_by_0 let p, q be XFinSequence of ; ::_thesis: ( p is dominated_by_0 & 2 * (Sum (p | k)) = k & p = (p | k) ^ q implies q is dominated_by_0 ) assume that A1: p is dominated_by_0 and A2: 2 * (Sum (p | k)) = k and A3: p = (p | k) ^ q ; ::_thesis: q is dominated_by_0 A4: len (p | k) = k by A1, A2, Th11; ( rng q c= rng p & rng p c= {0,1} ) by A1, A3, Def1, AFINSQ_1:25; hence rng q c= {0,1} by XBOOLE_1:1; :: according to CATALAN2:def_1 ::_thesis: for k being Nat st k <= dom q holds 2 * (Sum (q | k)) <= k let n be Nat; ::_thesis: ( n <= dom q implies 2 * (Sum (q | n)) <= n ) assume n <= dom q ; ::_thesis: 2 * (Sum (q | n)) <= n p | ((len (p | k)) + n) = (p | k) ^ (q | n) by A3, AFINSQ_1:59; then A5: Sum (p | ((len (p | k)) + n)) = (Sum (p | k)) + (Sum (q | n)) by AFINSQ_2:55; 2 * (Sum (p | ((len (p | k)) + n))) <= (len (p | k)) + n by A1, Th2; then k + (2 * (Sum (q | n))) <= (len (p | k)) + n by A2, A5; hence 2 * (Sum (q | n)) <= n by A4, XREAL_1:6; ::_thesis: verum end; theorem Th13: :: CATALAN2:13 for k, n being Nat for p being XFinSequence of st p is dominated_by_0 & 2 * (Sum (p | k)) = k & k = n + 1 holds p | k = (p | n) ^ (1 --> 1) proof let k, n be Nat; ::_thesis: for p being XFinSequence of st p is dominated_by_0 & 2 * (Sum (p | k)) = k & k = n + 1 holds p | k = (p | n) ^ (1 --> 1) let p be XFinSequence of ; ::_thesis: ( p is dominated_by_0 & 2 * (Sum (p | k)) = k & k = n + 1 implies p | k = (p | n) ^ (1 --> 1) ) assume that A1: p is dominated_by_0 and A2: 2 * (Sum (p | k)) = k and A3: k = n + 1 ; ::_thesis: p | k = (p | n) ^ (1 --> 1) reconsider q = p | k as XFinSequence of ; q . n = 1 proof Sum (p | k) <> 0 by A2, A3; then reconsider s = (Sum (p | k)) - 1 as Element of NAT by NAT_1:14, NAT_1:21; A4: q is dominated_by_0 by A1, Th6; then A5: rng q c= {0,1} by Def1; (2 * s) + 1 = n by A2, A3; then A6: ( Sum <%0%> = 0 & 2 * (Sum (q | n)) < n ) by A4, Th8, AFINSQ_2:53; A7: len q = n + 1 by A1, A2, A3, Th11; then A8: q = (q | n) ^ <%(q . n)%> by AFINSQ_1:56; n < n + 1 by NAT_1:13; then n in n + 1 by NAT_1:44; then A9: q . n in rng q by A7, FUNCT_1:3; assume q . n <> 1 ; ::_thesis: contradiction then q . n = 0 by A5, A9, TARSKI:def_2; then Sum q = (Sum (q | n)) + (Sum <%0%>) by A8, AFINSQ_2:55; hence contradiction by A2, A3, A6, NAT_1:13; ::_thesis: verum end; then A10: ( dom <%(q . n)%> = 1 & rng <%(q . n)%> = {1} ) by AFINSQ_1:33; n <= n + 1 by NAT_1:11; then n c= k by A3, NAT_1:39; then A11: q | n = p | n by RELAT_1:74; len q = n + 1 by A1, A2, A3, Th11; then q = (q | n) ^ <%(q . n)%> by AFINSQ_1:56; hence p | k = (p | n) ^ (1 --> 1) by A11, A10, FUNCOP_1:9; ::_thesis: verum end; theorem Th14: :: CATALAN2:14 for m being Nat for p being XFinSequence of st m = min* { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } & m > 0 & p is dominated_by_0 holds ex q being XFinSequence of st ( p | m = ((1 --> 0) ^ q) ^ (1 --> 1) & q is dominated_by_0 ) proof A1: ( dom <%0%> = 1 & rng <%0%> = {0} ) by AFINSQ_1:33; set q1 = 1 --> 1; set q0 = 1 --> 0; let m be Nat; ::_thesis: for p being XFinSequence of st m = min* { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } & m > 0 & p is dominated_by_0 holds ex q being XFinSequence of st ( p | m = ((1 --> 0) ^ q) ^ (1 --> 1) & q is dominated_by_0 ) let p be XFinSequence of ; ::_thesis: ( m = min* { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } & m > 0 & p is dominated_by_0 implies ex q being XFinSequence of st ( p | m = ((1 --> 0) ^ q) ^ (1 --> 1) & q is dominated_by_0 ) ) assume that A2: ( m = min* { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } & m > 0 ) and A3: p is dominated_by_0 ; ::_thesis: ex q being XFinSequence of st ( p | m = ((1 --> 0) ^ q) ^ (1 --> 1) & q is dominated_by_0 ) reconsider M = { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } as non empty Subset of NAT by A2, NAT_1:def_1; min* M in M by NAT_1:def_1; then consider n being Element of NAT such that A4: n = min* M and A5: 2 * (Sum (p | n)) = n and A6: n > 0 ; reconsider n1 = n - 1 as Element of NAT by A6, NAT_1:20; Sum (p | n) <> 0 by A5, A6; then n >= 2 * 1 by A5, NAT_1:14, XREAL_1:64; then A7: n1 >= 2 - 1 by XREAL_1:9; then A8: 1 c= n1 by NAT_1:39; then A9: (p | n1) | 1 = p | 1 by RELAT_1:74; A10: n1 < n1 + 1 by NAT_1:13; n <= len p by A3, A5, Th11; then A11: n1 < len p by A10, XXREAL_0:2; then 1 < len p by A7, XXREAL_0:2; then len (p | 1) = 1 by AFINSQ_1:11; then A12: p | 1 = <%((p | 1) . 0)%> by AFINSQ_1:34; p | 1 is dominated_by_0 by A3, Th6; then (p | 1) . 0 = 0 by Th3; then A13: p | 1 = 1 --> 0 by A12, A1, FUNCOP_1:9; consider q being XFinSequence of such that A14: p | n1 = ((p | n1) | 1) ^ q by Th1; set qq = ((1 --> 0) ^ q) ^ (1 --> 1); take q ; ::_thesis: ( p | m = ((1 --> 0) ^ q) ^ (1 --> 1) & q is dominated_by_0 ) A15: p | (n1 + 1) = (p | n1) ^ (1 --> 1) by A3, A5, Th13; hence p | m = ((1 --> 0) ^ q) ^ (1 --> 1) by A2, A4, A14, A8, A13, RELAT_1:74; ::_thesis: q is dominated_by_0 ( rng q c= rng ((1 --> 0) ^ q) & rng ((1 --> 0) ^ q) c= rng (((1 --> 0) ^ q) ^ (1 --> 1)) ) by AFINSQ_1:24, AFINSQ_1:25; then A16: rng q c= rng (((1 --> 0) ^ q) ^ (1 --> 1)) by XBOOLE_1:1; p | m is dominated_by_0 by A3, Th6; then rng (((1 --> 0) ^ q) ^ (1 --> 1)) c= {0,1} by A2, A4, A14, A13, A9, A15, Def1; hence rng q c= {0,1} by A16, XBOOLE_1:1; :: according to CATALAN2:def_1 ::_thesis: for k being Nat st k <= dom q holds 2 * (Sum (q | k)) <= k A17: dom (1 --> 0) = 1 by FUNCOP_1:13; len (p | n1) = n1 by A11, AFINSQ_1:11; then A18: n1 = (len (1 --> 0)) + (len q) by A14, A13, A9, AFINSQ_1:17; let k be Nat; ::_thesis: ( k <= dom q implies 2 * (Sum (q | k)) <= k ) assume k <= dom q ; ::_thesis: 2 * (Sum (q | k)) <= k then A19: (len (1 --> 0)) + k <= n1 by A18, XREAL_1:6; then (len (1 --> 0)) + k c= n1 by NAT_1:39; then A20: (p | n1) | (1 + k) = p | (1 + k) by A17, RELAT_1:74; A21: 1 + k < n by A15, A19, A17, NAT_1:13; A22: 2 * (Sum (p | (1 + k))) < 1 + k proof assume A23: 2 * (Sum (p | (1 + k))) >= 1 + k ; ::_thesis: contradiction 2 * (Sum (p | (k + 1))) <= k + 1 by A3, Th2; then 2 * (Sum (p | (1 + k))) = 1 + k by A23, XXREAL_0:1; then 1 + k in M ; hence contradiction by A4, A21, NAT_1:def_1; ::_thesis: verum end; (p | n1) | (1 + k) = (1 --> 0) ^ (q | k) by A14, A13, A9, A17, AFINSQ_1:59; then A24: Sum (p | (1 + k)) = (Sum (1 --> 0)) + (Sum (q | k)) by A20, AFINSQ_2:55; Sum (1 --> 0) = 0 * 1 by AFINSQ_2:58; hence 2 * (Sum (q | k)) <= k by A24, A22, NAT_1:13; ::_thesis: verum end; theorem Th15: :: CATALAN2:15 for p being XFinSequence of st rng p c= {0,1} & not p is dominated_by_0 holds ex k being Nat st ( (2 * k) + 1 = min* { N where N is Element of NAT : 2 * (Sum (p | N)) > N } & (2 * k) + 1 <= dom p & k = Sum (p | (2 * k)) & p . (2 * k) = 1 ) proof let p be XFinSequence of ; ::_thesis: ( rng p c= {0,1} & not p is dominated_by_0 implies ex k being Nat st ( (2 * k) + 1 = min* { N where N is Element of NAT : 2 * (Sum (p | N)) > N } & (2 * k) + 1 <= dom p & k = Sum (p | (2 * k)) & p . (2 * k) = 1 ) ) assume that A1: rng p c= {0,1} and A2: not p is dominated_by_0 ; ::_thesis: ex k being Nat st ( (2 * k) + 1 = min* { N where N is Element of NAT : 2 * (Sum (p | N)) > N } & (2 * k) + 1 <= dom p & k = Sum (p | (2 * k)) & p . (2 * k) = 1 ) set M = { N where N is Element of NAT : 2 * (Sum (p | N)) > N } ; { N where N is Element of NAT : 2 * (Sum (p | N)) > N } c= NAT proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { N where N is Element of NAT : 2 * (Sum (p | N)) > N } or x in NAT ) assume x in { N where N is Element of NAT : 2 * (Sum (p | N)) > N } ; ::_thesis: x in NAT then ex N being Element of NAT st ( x = N & 2 * (Sum (p | N)) > N ) ; hence x in NAT ; ::_thesis: verum end; then reconsider M = { N where N is Element of NAT : 2 * (Sum (p | N)) > N } as Subset of NAT ; consider k being Nat such that A3: k <= dom p and A4: 2 * (Sum (p | k)) > k by A1, A2, Def1; reconsider k = k as Element of NAT by ORDINAL1:def_12; k in M by A4; then reconsider M = M as non empty Subset of NAT ; min* M in M by NAT_1:def_1; then consider n being Element of NAT such that A5: min* M = n and A6: 2 * (Sum (p | n)) > n ; n > 0 by A6; then reconsider n1 = n - 1 as Element of NAT by NAT_1:20; reconsider S = Sum (p | n1) as Element of NAT by ORDINAL1:def_12; take S ; ::_thesis: ( (2 * S) + 1 = min* { N where N is Element of NAT : 2 * (Sum (p | N)) > N } & (2 * S) + 1 <= dom p & S = Sum (p | (2 * S)) & p . (2 * S) = 1 ) k in M by A4; then A7: k >= n by A5, NAT_1:def_1; then A8: dom p >= n by A3, XXREAL_0:2; A9: 2 * (Sum (p | n1)) = n1 proof A10: n1 < n1 + 1 by NAT_1:13; then n1 c= n1 + 1 by NAT_1:39; then A11: (p | n) | n1 = p | n1 by RELAT_1:74; ( ( n = len p & p | (dom p) = p ) or n < len p ) by A8, XXREAL_0:1; then A12: len (p | n) = n1 + 1 by AFINSQ_1:11; then n1 in len (p | n) by A10, NAT_1:44; then A13: (p | n) . n1 in rng (p | n) by FUNCT_1:3; p | n = ((p | n) | n1) ^ <%((p | n) . n1)%> by A12, AFINSQ_1:56; then Sum (p | n) = (Sum (p | n1)) + (Sum <%((p | n) . n1)%>) by A11, AFINSQ_2:55; then A14: (2 * (Sum (p | n1))) + (2 * (Sum <%((p | n) . n1)%>)) >= n + 1 by A6, NAT_1:13; n1 < n1 + 1 by NAT_1:13; then not n1 in M by A5, NAT_1:def_1; then A15: 2 * (Sum (p | n1)) <= n1 ; rng (p | n) c= rng p by RELAT_1:70; then (p | n) . n1 in {0,1} by A1, A13, TARSKI:def_3; then A16: ( (p | n) . n1 = 0 or (p | n) . n1 = 1 ) by TARSKI:def_2; assume 2 * (Sum (p | n1)) <> n1 ; ::_thesis: contradiction then ( Sum <%((p | n) . n1)%> = (p | n) . n1 & 2 * (Sum (p | n1)) < n1 ) by A15, AFINSQ_2:53, XXREAL_0:1; then (2 * (Sum (p | n1))) + (2 * (Sum <%((p | n) . n1)%>)) < n1 + 2 by A16, XREAL_1:8; hence contradiction by A14; ::_thesis: verum end; p . n1 = 1 proof n c= len p by A8, NAT_1:39; then A17: dom (p | n) = n1 + 1 by RELAT_1:62; dom (p | n) = len (p | n) ; then A18: ( Sum <%0%> = 0 & p | n = ((p | n) | n1) ^ <%((p | n) . n1)%> ) by A17, AFINSQ_1:56, AFINSQ_2:53; assume A19: p . n1 <> 1 ; ::_thesis: contradiction A20: n1 < n1 + 1 by NAT_1:13; then n1 < dom p by A8, XXREAL_0:2; then A21: n1 in dom p by NAT_1:44; n1 c= n by A20, NAT_1:39; then A22: (p | n) | n1 = p | n1 by RELAT_1:74; n1 in n by A20, NAT_1:44; then n1 in (dom p) /\ n by A21, XBOOLE_0:def_4; then A23: (p | n) . n1 = p . n1 by FUNCT_1:48; A24: n1 < n1 + 1 by NAT_1:13; p . n1 in rng p by A21, FUNCT_1:3; then p . n1 = 0 by A1, A19, TARSKI:def_2; then Sum (p | n) = (Sum (p | n1)) + 0 by A18, A23, A22, AFINSQ_2:55; hence contradiction by A6, A9, A24; ::_thesis: verum end; hence ( (2 * S) + 1 = min* { N where N is Element of NAT : 2 * (Sum (p | N)) > N } & (2 * S) + 1 <= dom p & S = Sum (p | (2 * S)) & p . (2 * S) = 1 ) by A3, A5, A7, A9, XXREAL_0:2; ::_thesis: verum end; theorem Th16: :: CATALAN2:16 for p, q being XFinSequence of for k being Nat st p | ((2 * k) + (len q)) = ((k --> 0) ^ q) ^ (k --> 1) & q is dominated_by_0 & 2 * (Sum q) = len q & k > 0 holds min* { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } = (2 * k) + (len q) proof let p, q be XFinSequence of ; ::_thesis: for k being Nat st p | ((2 * k) + (len q)) = ((k --> 0) ^ q) ^ (k --> 1) & q is dominated_by_0 & 2 * (Sum q) = len q & k > 0 holds min* { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } = (2 * k) + (len q) let k be Nat; ::_thesis: ( p | ((2 * k) + (len q)) = ((k --> 0) ^ q) ^ (k --> 1) & q is dominated_by_0 & 2 * (Sum q) = len q & k > 0 implies min* { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } = (2 * k) + (len q) ) assume that A1: p | ((2 * k) + (len q)) = ((k --> 0) ^ q) ^ (k --> 1) and A2: q is dominated_by_0 and A3: 2 * (Sum q) = len q and A4: k > 0 ; ::_thesis: min* { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } = (2 * k) + (len q) set k0 = k --> 0; A5: Sum (k --> 0) = k * 0 by AFINSQ_2:58; then A6: 2 * k > 0 by A4, XREAL_1:68; reconsider k1 = k --> 1 as XFinSequence of ; set M = { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } ; set kqk = ((k --> 0) ^ q) ^ k1; Sum (((k --> 0) ^ q) ^ k1) = (Sum ((k --> 0) ^ q)) + (Sum k1) by AFINSQ_2:55; then A7: Sum (((k --> 0) ^ q) ^ k1) = ((Sum (k --> 0)) + (Sum q)) + (Sum k1) by AFINSQ_2:55; Sum k1 = k * 1 by AFINSQ_2:58; then 2 * (Sum (p | ((2 * k) + (len q)))) = (len q) + (2 * k) by A1, A3, A7, A5; then A8: (2 * k) + (len q) in { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } by A6; { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } c= NAT proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } or y in NAT ) assume y in { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } ; ::_thesis: y in NAT then ex i being Element of NAT st ( i = y & 2 * (Sum (p | i)) = i & i > 0 ) ; hence y in NAT ; ::_thesis: verum end; then reconsider M = { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } as non empty Subset of NAT by A8; min* M = (2 * k) + (len q) proof ((k --> 0) ^ q) ^ k1 = (k --> 0) ^ (q ^ k1) by AFINSQ_1:27; then A9: len (((k --> 0) ^ q) ^ k1) = (len (k --> 0)) + (len (q ^ k1)) by AFINSQ_1:17; dom (k --> 0) = k by FUNCOP_1:13; then A10: len (((k --> 0) ^ q) ^ k1) = k + ((len q) + (len k1)) by A9, AFINSQ_1:17; assume A11: min* M <> (2 * k) + (len q) ; ::_thesis: contradiction min* M in M by NAT_1:def_1; then A12: ex i being Element of NAT st ( i = min* M & 2 * (Sum (p | i)) = i & i > 0 ) ; A13: dom k1 = k by FUNCOP_1:13; A14: (2 * k) + (len q) >= min* M by A8, NAT_1:def_1; then A15: min* M c= (2 * k) + (len q) by NAT_1:39; then A16: p | (min* M) = (((k --> 0) ^ q) ^ k1) | (min* M) by A1, RELAT_1:74; now__::_thesis:_contradiction percases ( min* M <= k or min* M > k ) ; supposeA17: min* M <= k ; ::_thesis: contradiction ( k = dom (k --> 0) & ((k --> 0) ^ q) ^ k1 = (k --> 0) ^ (q ^ k1) ) by AFINSQ_1:27, FUNCOP_1:13; then A18: (((k --> 0) ^ q) ^ k1) | (min* M) = (k --> 0) | (min* M) by A17, AFINSQ_1:58; A19: Sum ((min* M) --> 0) = (min* M) * 0 by AFINSQ_2:58; (k --> 0) | (min* M) = (min* M) --> 0 by A17, Lm1; then Sum (p | (min* M)) = Sum ((min* M) --> 0) by A1, A15, A18, RELAT_1:74; hence contradiction by A12, A19; ::_thesis: verum end; suppose min* M > k ; ::_thesis: contradiction then reconsider mk = (min* M) - k as Element of NAT by NAT_1:21; now__::_thesis:_contradiction percases ( min* M <= k + (len q) or min* M > k + (len q) ) ; supposeA20: min* M <= k + (len q) ; ::_thesis: contradiction A21: dom (k --> 0) = k by FUNCOP_1:13; min* M = mk + k ; then A22: ((k --> 0) ^ q) | (min* M) = (k --> 0) ^ (q | mk) by A21, AFINSQ_1:59; dom ((k --> 0) ^ q) = (len (k --> 0)) + (len q) by AFINSQ_1:def_3; then (((k --> 0) ^ q) ^ k1) | (min* M) = ((k --> 0) ^ q) | (min* M) by A20, A21, AFINSQ_1:58; then A23: Sum (p | (min* M)) = (Sum (k --> 0)) + (Sum (q | mk)) by A16, A22, AFINSQ_2:55; A24: 1 <= k by A4, NAT_1:14; Sum (k --> 0) = k * 0 by AFINSQ_2:58; then mk + k <= mk by A2, A12, A23, Th2; hence contradiction by A24, NAT_1:19; ::_thesis: verum end; suppose min* M > k + (len q) ; ::_thesis: contradiction then reconsider mkL = (min* M) - (k + (len q)) as Element of NAT by NAT_1:21; A25: 2 * (Sum (p | (min* M))) = min* M by A12; ( dom ((k --> 0) ^ q) = (len (k --> 0)) + (len q) & dom (k --> 0) = k ) by AFINSQ_1:def_3, FUNCOP_1:13; then min* M = (dom ((k --> 0) ^ q)) + mkL ; then (((k --> 0) ^ q) ^ k1) | (min* M) = ((k --> 0) ^ q) ^ (k1 | mkL) by AFINSQ_1:59; then A26: Sum (p | (min* M)) = (Sum ((k --> 0) ^ q)) + (Sum (k1 | mkL)) by A16, AFINSQ_2:55; min* M < len (((k --> 0) ^ q) ^ k1) by A11, A10, A13, A14, XXREAL_0:1; then mkL < ((2 * k) + (len q)) - (k + (len q)) by A10, A13, XREAL_1:9; then k1 | mkL = mkL --> 1 by Lm1; then A27: Sum (k1 | mkL) = mkL * 1 by AFINSQ_2:58; ( Sum ((k --> 0) ^ q) = (Sum (k --> 0)) + (Sum q) & Sum (k --> 0) = k * 0 ) by AFINSQ_2:55, AFINSQ_2:58; hence contradiction by A3, A11, A26, A27, A25; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; hence min* { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } = (2 * k) + (len q) ; ::_thesis: verum end; theorem Th17: :: CATALAN2:17 for p being XFinSequence of st p is dominated_by_0 & { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } = {} & len p > 0 holds ex q being XFinSequence of st ( p = <%0%> ^ q & q is dominated_by_0 ) proof let p be XFinSequence of ; ::_thesis: ( p is dominated_by_0 & { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } = {} & len p > 0 implies ex q being XFinSequence of st ( p = <%0%> ^ q & q is dominated_by_0 ) ) assume that A1: p is dominated_by_0 and A2: ( { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } = {} & len p > 0 ) ; ::_thesis: ex q being XFinSequence of st ( p = <%0%> ^ q & q is dominated_by_0 ) set M = { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } ; consider q being XFinSequence of such that A3: p = (p | 1) ^ q by Th1; take q ; ::_thesis: ( p = <%0%> ^ q & q is dominated_by_0 ) A4: rng p c= {0,1} by A1, Def1; rng q c= rng p by A3, AFINSQ_1:25; then A5: rng q c= {0,1} by A4, XBOOLE_1:1; len p >= 1 by A2, NAT_1:14; then 1 c= dom p by NAT_1:39; then A6: dom (p | 1) = 1 by RELAT_1:62; len (p | 1) = dom (p | 1) ; then A7: p | 1 = <%((p | 1) . 0)%> by A6, AFINSQ_1:34; 0 in 1 by NAT_1:44; then A8: (p | 1) . 0 = p . 0 by A6, FUNCT_1:47; hence p = <%0%> ^ q by A1, A3, A7, Th3; ::_thesis: q is dominated_by_0 assume not q is dominated_by_0 ; ::_thesis: contradiction then consider i being Nat such that i <= dom q and A9: 2 * (Sum (q | i)) > i by A5, Def1; reconsider i = i as Element of NAT by ORDINAL1:def_12; p | (1 + i) = (p | 1) ^ (q | i) by A3, A6, AFINSQ_1:59; then A10: Sum (p | (1 + i)) = (Sum <%(p . 0)%>) + (Sum (q | i)) by A7, A8, AFINSQ_2:55; A11: 2 * (Sum (q | i)) >= i + 1 by A9, NAT_1:13; Sum <%(p . 0)%> = p . 0 by AFINSQ_2:53; then A12: Sum (p | (1 + i)) = 0 + (Sum (q | i)) by A1, A10, Th3; then 1 + i >= 2 * (Sum (q | i)) by A1, Th2; then 1 + i = 2 * (Sum (q | i)) by A11, XXREAL_0:1; then 1 + i in { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } by A12; hence contradiction by A2; ::_thesis: verum end; theorem Th18: :: CATALAN2:18 for p being XFinSequence of st p is dominated_by_0 holds ( <%0%> ^ p is dominated_by_0 & { N where N is Element of NAT : ( 2 * (Sum ((<%0%> ^ p) | N)) = N & N > 0 ) } = {} ) proof let p be XFinSequence of ; ::_thesis: ( p is dominated_by_0 implies ( <%0%> ^ p is dominated_by_0 & { N where N is Element of NAT : ( 2 * (Sum ((<%0%> ^ p) | N)) = N & N > 0 ) } = {} ) ) reconsider q = 1 --> 0 as XFinSequence of ; assume A1: p is dominated_by_0 ; ::_thesis: ( <%0%> ^ p is dominated_by_0 & { N where N is Element of NAT : ( 2 * (Sum ((<%0%> ^ p) | N)) = N & N > 0 ) } = {} ) ( dom q = 1 & len q = dom q ) by FUNCOP_1:13; then A2: q = <%(q . 0)%> by AFINSQ_1:34; q is dominated_by_0 by Lm2; then ( q is dominated_by_0 & q . 0 = 0 ) by Th3; hence <%0%> ^ p is dominated_by_0 by A1, A2, Th7; ::_thesis: { N where N is Element of NAT : ( 2 * (Sum ((<%0%> ^ p) | N)) = N & N > 0 ) } = {} set M = { N where N is Element of NAT : ( 2 * (Sum ((<%0%> ^ p) | N)) = N & N > 0 ) } ; assume { N where N is Element of NAT : ( 2 * (Sum ((<%0%> ^ p) | N)) = N & N > 0 ) } <> {} ; ::_thesis: contradiction then consider x being set such that A3: x in { N where N is Element of NAT : ( 2 * (Sum ((<%0%> ^ p) | N)) = N & N > 0 ) } by XBOOLE_0:def_1; consider i being Element of NAT such that x = i and A4: 2 * (Sum ((<%0%> ^ p) | i)) = i and A5: i > 0 by A3; reconsider i1 = i - 1 as Element of NAT by A5, NAT_1:20; dom <%0%> = 1 by AFINSQ_1:33; then i = (dom <%0%>) + i1 ; then (<%0%> ^ p) | i = <%0%> ^ (p | i1) by AFINSQ_1:59; then A6: Sum ((<%0%> ^ p) | i) = (Sum <%0%>) + (Sum (p | i1)) by AFINSQ_2:55; ( Sum <%0%> = 0 & i1 < i1 + 1 ) by AFINSQ_2:53, NAT_1:13; hence contradiction by A1, A4, A6, Th2; ::_thesis: verum end; theorem :: CATALAN2:19 for k being Nat for p being XFinSequence of st rng p c= {0,1} & (2 * k) + 1 = min* { N where N is Element of NAT : 2 * (Sum (p | N)) > N } holds p | (2 * k) is dominated_by_0 proof let k be Nat; ::_thesis: for p being XFinSequence of st rng p c= {0,1} & (2 * k) + 1 = min* { N where N is Element of NAT : 2 * (Sum (p | N)) > N } holds p | (2 * k) is dominated_by_0 let p be XFinSequence of ; ::_thesis: ( rng p c= {0,1} & (2 * k) + 1 = min* { N where N is Element of NAT : 2 * (Sum (p | N)) > N } implies p | (2 * k) is dominated_by_0 ) set M = { N where N is Element of NAT : 2 * (Sum (p | N)) > N } ; set q = p | (2 * k); assume that A1: rng p c= {0,1} and A2: (2 * k) + 1 = min* { N where N is Element of NAT : 2 * (Sum (p | N)) > N } ; ::_thesis: p | (2 * k) is dominated_by_0 rng (p | (2 * k)) c= rng p by RELAT_1:70; hence rng (p | (2 * k)) c= {0,1} by A1, XBOOLE_1:1; :: according to CATALAN2:def_1 ::_thesis: for k being Nat st k <= dom (p | (2 * k)) holds 2 * (Sum ((p | (2 * k)) | k)) <= k reconsider M = { N where N is Element of NAT : 2 * (Sum (p | N)) > N } as non empty Subset of NAT by A2, NAT_1:def_1; let m be Nat; ::_thesis: ( m <= dom (p | (2 * k)) implies 2 * (Sum ((p | (2 * k)) | m)) <= m ) assume m <= dom (p | (2 * k)) ; ::_thesis: 2 * (Sum ((p | (2 * k)) | m)) <= m then A3: m c= dom (p | (2 * k)) by NAT_1:39; m c= 2 * k by A3, XBOOLE_1:1; then m <= 2 * k by NAT_1:39; then A4: m < (2 * k) + 1 by NAT_1:13; assume A5: 2 * (Sum ((p | (2 * k)) | m)) > m ; ::_thesis: contradiction reconsider m = m as Element of NAT by ORDINAL1:def_12; ( (p | (2 * k)) | m = p | m & m in NAT ) by A3, RELAT_1:74, XBOOLE_1:1; then m in M by A5; hence contradiction by A2, A4, NAT_1:def_1; ::_thesis: verum end; begin definition let n, m be Nat; func Domin_0 (n,m) -> Subset of ({0,1} ^omega) means :Def2: :: CATALAN2:def 2 for x being set holds ( x in it iff ex p being XFinSequence of st ( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ); existence ex b1 being Subset of ({0,1} ^omega) st for x being set holds ( x in b1 iff ex p being XFinSequence of st ( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ) proof defpred S1[ set ] means ex p being XFinSequence of st ( p = $1 & p is dominated_by_0 & dom p = n & Sum p = m ); consider X being set such that A1: for x being set holds ( x in X iff ( x in bool [:NAT,NAT:] & S1[x] ) ) from XBOOLE_0:sch_1(); X c= {0,1} ^omega proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in {0,1} ^omega ) assume x in X ; ::_thesis: x in {0,1} ^omega then consider p being XFinSequence of such that A2: p = x and A3: p is dominated_by_0 and dom p = n and Sum p = m by A1; rng p c= {0,1} by A3, Def1; then p is XFinSequence of by RELAT_1:def_19; hence x in {0,1} ^omega by A2, AFINSQ_1:42; ::_thesis: verum end; then reconsider X = X as Subset of ({0,1} ^omega) ; take X ; ::_thesis: for x being set holds ( x in X iff ex p being XFinSequence of st ( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ) let x be set ; ::_thesis: ( x in X iff ex p being XFinSequence of st ( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ) thus ( x in X implies S1[x] ) by A1; ::_thesis: ( ex p being XFinSequence of st ( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) implies x in X ) given p being XFinSequence of such that A4: ( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ; ::_thesis: x in X ( p c= [:(dom p),(rng p):] & [:(dom p),(rng p):] c= [:NAT,NAT:] ) by RELAT_1:7, ZFMISC_1:96; then p c= [:NAT,NAT:] by XBOOLE_1:1; hence x in X by A1, A4; ::_thesis: verum end; uniqueness for b1, b2 being Subset of ({0,1} ^omega) st ( for x being set holds ( x in b1 iff ex p being XFinSequence of st ( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ) ) & ( for x being set holds ( x in b2 iff ex p being XFinSequence of st ( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ) ) holds b1 = b2 proof let X1, X2 be Subset of ({0,1} ^omega); ::_thesis: ( ( for x being set holds ( x in X1 iff ex p being XFinSequence of st ( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ) ) & ( for x being set holds ( x in X2 iff ex p being XFinSequence of st ( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ) ) implies X1 = X2 ) assume that A5: for x being set holds ( x in X1 iff ex p being XFinSequence of st ( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ) and A6: for x being set holds ( x in X2 iff ex p being XFinSequence of st ( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ) ; ::_thesis: X1 = X2 for x being set holds ( x in X1 iff x in X2 ) proof let x be set ; ::_thesis: ( x in X1 iff x in X2 ) ( x in X1 iff ex p being XFinSequence of st ( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ) by A5; hence ( x in X1 iff x in X2 ) by A6; ::_thesis: verum end; hence X1 = X2 by TARSKI:1; ::_thesis: verum end; end; :: deftheorem Def2 defines Domin_0 CATALAN2:def_2_:_ for n, m being Nat for b3 being Subset of ({0,1} ^omega) holds ( b3 = Domin_0 (n,m) iff for x being set holds ( x in b3 iff ex p being XFinSequence of st ( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ) ); theorem Th20: :: CATALAN2:20 for n, m being Nat for p being XFinSequence of holds ( p in Domin_0 (n,m) iff ( p is dominated_by_0 & dom p = n & Sum p = m ) ) proof let n, m be Nat; ::_thesis: for p being XFinSequence of holds ( p in Domin_0 (n,m) iff ( p is dominated_by_0 & dom p = n & Sum p = m ) ) let p be XFinSequence of ; ::_thesis: ( p in Domin_0 (n,m) iff ( p is dominated_by_0 & dom p = n & Sum p = m ) ) thus ( p in Domin_0 (n,m) implies ( p is dominated_by_0 & dom p = n & Sum p = m ) ) ::_thesis: ( p is dominated_by_0 & dom p = n & Sum p = m implies p in Domin_0 (n,m) ) proof assume p in Domin_0 (n,m) ; ::_thesis: ( p is dominated_by_0 & dom p = n & Sum p = m ) then ex q being XFinSequence of st ( q = p & q is dominated_by_0 & dom q = n & Sum q = m ) by Def2; hence ( p is dominated_by_0 & dom p = n & Sum p = m ) ; ::_thesis: verum end; thus ( p is dominated_by_0 & dom p = n & Sum p = m implies p in Domin_0 (n,m) ) by Def2; ::_thesis: verum end; theorem Th21: :: CATALAN2:21 for n, m being Nat holds Domin_0 (n,m) c= Choose (n,m,1,0) proof let n, m be Nat; ::_thesis: Domin_0 (n,m) c= Choose (n,m,1,0) let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Domin_0 (n,m) or x in Choose (n,m,1,0) ) assume x in Domin_0 (n,m) ; ::_thesis: x in Choose (n,m,1,0) then consider p being XFinSequence of such that A1: p = x and A2: p is dominated_by_0 and A3: ( dom p = n & Sum p = m ) by Def2; rng p c= {0,1} by A2, Def1; hence x in Choose (n,m,1,0) by A1, A3, CARD_FIN:40; ::_thesis: verum end; registration let n, m be Nat; cluster Domin_0 (n,m) -> finite ; coherence Domin_0 (n,m) is finite proof Domin_0 (n,m) c= Choose (n,m,1,0) by Th21; hence Domin_0 (n,m) is finite ; ::_thesis: verum end; end; theorem Th22: :: CATALAN2:22 for n, m being Nat holds ( Domin_0 (n,m) is empty iff 2 * m > n ) proof let n, m be Nat; ::_thesis: ( Domin_0 (n,m) is empty iff 2 * m > n ) thus ( Domin_0 (n,m) is empty implies 2 * m > n ) ::_thesis: ( 2 * m > n implies Domin_0 (n,m) is empty ) proof set q = m --> 1; assume A1: Domin_0 (n,m) is empty ; ::_thesis: 2 * m > n assume A2: 2 * m <= n ; ::_thesis: contradiction m <= m + m by NAT_1:12; then reconsider nm = n - m as Element of NAT by A2, NAT_1:21, XXREAL_0:2; set p = nm --> 0; (2 * m) - m <= nm by A2, XREAL_1:9; then A3: (nm --> 0) ^ (m --> 1) is dominated_by_0 by Th5; ( dom ((nm --> 0) ^ (m --> 1)) = (len (nm --> 0)) + (len (m --> 1)) & dom (nm --> 0) = nm ) by AFINSQ_1:def_3, FUNCOP_1:13; then A4: dom ((nm --> 0) ^ (m --> 1)) = nm + m by FUNCOP_1:13; A5: Sum ((nm --> 0) ^ (m --> 1)) = (Sum (nm --> 0)) + (Sum (m --> 1)) by AFINSQ_2:55; ( Sum (nm --> 0) = 0 * nm & Sum (m --> 1) = 1 * m ) by AFINSQ_2:58; hence contradiction by A1, A5, A4, A3, Def2; ::_thesis: verum end; assume A6: 2 * m > n ; ::_thesis: Domin_0 (n,m) is empty assume not Domin_0 (n,m) is empty ; ::_thesis: contradiction then consider x being set such that A7: x in Domin_0 (n,m) by XBOOLE_0:def_1; consider p being XFinSequence of such that p = x and A8: p is dominated_by_0 and A9: dom p = n and A10: Sum p = m by A7, Def2; p | n = p by A9, RELAT_1:69; hence contradiction by A6, A8, A10, Th2; ::_thesis: verum end; theorem Th23: :: CATALAN2:23 for n being Nat holds Domin_0 (n,0) = {(n --> 0)} proof let n be Nat; ::_thesis: Domin_0 (n,0) = {(n --> 0)} set p = n --> 0; A1: Domin_0 (n,0) c= {(n --> 0)} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Domin_0 (n,0) or x in {(n --> 0)} ) assume x in Domin_0 (n,0) ; ::_thesis: x in {(n --> 0)} then consider q being XFinSequence of such that A2: x = q and q is dominated_by_0 and A3: dom q = n and A4: Sum q = 0 by Def2; ( len q = n & q is nonnegative-yielding ) by A3; then ( q = n --> 0 or ( q = {} & n = 0 ) ) by A4, AFINSQ_2:62; then q = n --> 0 ; hence x in {(n --> 0)} by A2, TARSKI:def_1; ::_thesis: verum end; {(n --> 0)} c= Domin_0 (n,0) proof A5: n --> 0 is dominated_by_0 by Lm2; ( dom (n --> 0) = n & Sum (n --> 0) = n * 0 ) by AFINSQ_2:58, FUNCOP_1:13; then A6: n --> 0 in Domin_0 (n,0) by A5, Def2; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(n --> 0)} or x in Domin_0 (n,0) ) assume x in {(n --> 0)} ; ::_thesis: x in Domin_0 (n,0) hence x in Domin_0 (n,0) by A6, TARSKI:def_1; ::_thesis: verum end; hence Domin_0 (n,0) = {(n --> 0)} by A1, XBOOLE_0:def_10; ::_thesis: verum end; theorem Th24: :: CATALAN2:24 for n being Nat holds card (Domin_0 (n,0)) = 1 proof let n be Nat; ::_thesis: card (Domin_0 (n,0)) = 1 Domin_0 (n,0) = {(n --> 0)} by Th23; hence card (Domin_0 (n,0)) = 1 by CARD_1:30; ::_thesis: verum end; theorem Th25: :: CATALAN2:25 for p being XFinSequence of for n being Nat st rng p c= {0,n} holds ex q being XFinSequence of st ( len p = len q & rng q c= {0,n} & ( for k being Nat st k <= len p holds (Sum (p | k)) + (Sum (q | k)) = n * k ) & ( for k being Nat st k in len p holds q . k = n - (p . k) ) ) proof let p be XFinSequence of ; ::_thesis: for n being Nat st rng p c= {0,n} holds ex q being XFinSequence of st ( len p = len q & rng q c= {0,n} & ( for k being Nat st k <= len p holds (Sum (p | k)) + (Sum (q | k)) = n * k ) & ( for k being Nat st k in len p holds q . k = n - (p . k) ) ) let n be Nat; ::_thesis: ( rng p c= {0,n} implies ex q being XFinSequence of st ( len p = len q & rng q c= {0,n} & ( for k being Nat st k <= len p holds (Sum (p | k)) + (Sum (q | k)) = n * k ) & ( for k being Nat st k in len p holds q . k = n - (p . k) ) ) ) assume A1: rng p c= {0,n} ; ::_thesis: ex q being XFinSequence of st ( len p = len q & rng q c= {0,n} & ( for k being Nat st k <= len p holds (Sum (p | k)) + (Sum (q | k)) = n * k ) & ( for k being Nat st k in len p holds q . k = n - (p . k) ) ) reconsider nn = n as Element of NAT by ORDINAL1:def_12; defpred S1[ set , set ] means for k being Nat st k = $1 holds $2 = n - (p . k); A2: for k being Nat st k in len p holds ex x being Element of {0,n} st S1[k,x] proof let k be Nat; ::_thesis: ( k in len p implies ex x being Element of {0,n} st S1[k,x] ) assume k in len p ; ::_thesis: ex x being Element of {0,n} st S1[k,x] then p . k in rng p by FUNCT_1:3; then ( p . k = 0 or p . k = n ) by A1, TARSKI:def_2; then A3: n - (p . k) in {0,n} by TARSKI:def_2; S1[k,n - (p . k)] ; hence ex x being Element of {0,n} st S1[k,x] by A3; ::_thesis: verum end; consider q being XFinSequence of such that A4: ( dom q = len p & ( for k being Nat st k in len p holds S1[k,q . k] ) ) from STIRL2_1:sch_5(A2); rng q c= {0,nn} ; then rng q c= NAT by XBOOLE_1:1; then reconsider q = q as XFinSequence of by RELAT_1:def_19; defpred S2[ Nat] means ( $1 <= len p implies (Sum (p | $1)) + (Sum (q | $1)) = n * $1 ); A5: for k being Nat st S2[k] holds S2[k + 1] proof let k be Nat; ::_thesis: ( S2[k] implies S2[k + 1] ) assume A6: S2[k] ; ::_thesis: S2[k + 1] set k1 = k + 1; A7: k < k + 1 by NAT_1:13; then A8: k c= k + 1 by NAT_1:39; then A9: (p | (k + 1)) | k = p | k by RELAT_1:74; A10: (q | (k + 1)) | k = q | k by A8, RELAT_1:74; assume A11: k + 1 <= len p ; ::_thesis: (Sum (p | (k + 1))) + (Sum (q | (k + 1))) = n * (k + 1) then A12: k + 1 c= len p by NAT_1:39; then A13: len (q | (k + 1)) = k + 1 by A4, RELAT_1:62; then A14: q | (k + 1) = ((q | (k + 1)) | k) ^ <%((q | (k + 1)) . k)%> by AFINSQ_1:56; dom (p | (k + 1)) = k + 1 by A12, RELAT_1:62; then A15: k in dom (p | (k + 1)) by A7, NAT_1:44; then A16: (p | (k + 1)) . k = p . k by FUNCT_1:47; len (p | (k + 1)) = k + 1 by A12, RELAT_1:62; then p | (k + 1) = ((p | (k + 1)) | k) ^ <%((p | (k + 1)) . k)%> by AFINSQ_1:56; then Sum (p | (k + 1)) = (Sum (p | k)) + (Sum <%(p . k)%>) by A16, A9, AFINSQ_2:55; then A17: Sum (p | (k + 1)) = (Sum (p | k)) + (p . k) by AFINSQ_2:53; k < len p by A11, NAT_1:13; then k in len p by NAT_1:44; then A18: q . k = n - (p . k) by A4; (q | (k + 1)) . k = q . k by A13, A15, FUNCT_1:47; then Sum (q | (k + 1)) = (Sum (q | k)) + (Sum <%(q . k)%>) by A14, A10, AFINSQ_2:55; then Sum (q | (k + 1)) = (Sum (q | k)) + (n - (p . k)) by A18, AFINSQ_2:53; hence (Sum (p | (k + 1))) + (Sum (q | (k + 1))) = n * (k + 1) by A6, A11, A17, NAT_1:13; ::_thesis: verum end; take q ; ::_thesis: ( len p = len q & rng q c= {0,n} & ( for k being Nat st k <= len p holds (Sum (p | k)) + (Sum (q | k)) = n * k ) & ( for k being Nat st k in len p holds q . k = n - (p . k) ) ) thus len p = len q by A4; ::_thesis: ( rng q c= {0,n} & ( for k being Nat st k <= len p holds (Sum (p | k)) + (Sum (q | k)) = n * k ) & ( for k being Nat st k in len p holds q . k = n - (p . k) ) ) thus rng q c= {0,n} by RELAT_1:def_19; ::_thesis: ( ( for k being Nat st k <= len p holds (Sum (p | k)) + (Sum (q | k)) = n * k ) & ( for k being Nat st k in len p holds q . k = n - (p . k) ) ) A19: S2[ 0 ] ; for k being Nat holds S2[k] from NAT_1:sch_2(A19, A5); hence ( ( for k being Nat st k <= len p holds (Sum (p | k)) + (Sum (q | k)) = n * k ) & ( for k being Nat st k in len p holds q . k = n - (p . k) ) ) by A4; ::_thesis: verum end; theorem Th26: :: CATALAN2:26 for m, n being Nat st m <= n holds n choose m > 0 proof let m, n be Nat; ::_thesis: ( m <= n implies n choose m > 0 ) assume A1: m <= n ; ::_thesis: n choose m > 0 then reconsider nm = n - m as Nat by NAT_1:21; A2: (m !) * (nm !) > (m !) * 0 by XREAL_1:68; n ! > 0 * ((m !) * (nm !)) ; then (n !) / ((m !) * (nm !)) > 0 by A2, XREAL_1:81; hence n choose m > 0 by A1, NEWTON:def_3; ::_thesis: verum end; theorem Th27: :: CATALAN2:27 for m, n being Nat st 2 * (m + 1) <= n holds card ((Choose (n,(m + 1),1,0)) \ (Domin_0 (n,(m + 1)))) = card (Choose (n,m,1,0)) proof let m, n be Nat; ::_thesis: ( 2 * (m + 1) <= n implies card ((Choose (n,(m + 1),1,0)) \ (Domin_0 (n,(m + 1)))) = card (Choose (n,m,1,0)) ) defpred S1[ set , set ] means for p being XFinSequence of for k being Nat st $1 = p & (2 * k) + 1 = min* { N where N is Element of NAT : 2 * (Sum (p | N)) > N } holds ex r1, r2 being XFinSequence of st ( $2 = r1 ^ r2 & len r1 = (2 * k) + 1 & len r1 = len (p | ((2 * k) + 1)) & p = (p | ((2 * k) + 1)) ^ r2 & ( for o being Nat st o < (2 * k) + 1 holds r1 . o = 1 - (p . o) ) ); assume A1: 2 * (m + 1) <= n ; ::_thesis: card ((Choose (n,(m + 1),1,0)) \ (Domin_0 (n,(m + 1)))) = card (Choose (n,m,1,0)) A2: card n = n by CARD_1:def_2; A3: m <= m + m by XREAL_1:31; m <= m + 1 by NAT_1:13; then 2 * m <= 2 * (m + 1) by XREAL_1:64; then 2 * m <= n by A1, XXREAL_0:2; then m <= n by A3, XXREAL_0:2; then (card n) choose m > 0 by A2, Th26; then reconsider W = Choose (n,m,1,0) as non empty finite set by CARD_1:27, CARD_FIN:16; set Z = Domin_0 (n,(m + 1)); set CH = Choose (n,(m + 1),1,0); A4: for x being set st x in (Choose (n,(m + 1),1,0)) \ (Domin_0 (n,(m + 1))) holds ex y being set st ( y in W & S1[x,y] ) proof let x be set ; ::_thesis: ( x in (Choose (n,(m + 1),1,0)) \ (Domin_0 (n,(m + 1))) implies ex y being set st ( y in W & S1[x,y] ) ) assume A5: x in (Choose (n,(m + 1),1,0)) \ (Domin_0 (n,(m + 1))) ; ::_thesis: ex y being set st ( y in W & S1[x,y] ) x in Choose (n,(m + 1),1,0) by A5, XBOOLE_0:def_5; then consider p being XFinSequence of such that A6: p = x and A7: dom p = n and A8: rng p c= {0,1} and A9: Sum p = m + 1 by CARD_FIN:40; not p in Domin_0 (n,(m + 1)) by A5, A6, XBOOLE_0:def_5; then not p is dominated_by_0 by A7, A9, Def2; then consider o being Nat such that A10: ( (2 * o) + 1 = min* { N where N is Element of NAT : 2 * (Sum (p | N)) > N } & (2 * o) + 1 <= dom p & o = Sum (p | (2 * o)) & p . (2 * o) = 1 ) by A8, Th15; set q = p | ((2 * o) + 1); consider r2 being XFinSequence of such that A11: p = (p | ((2 * o) + 1)) ^ r2 by Th1; rng (p | ((2 * o) + 1)) c= rng p by RELAT_1:70; then rng (p | ((2 * o) + 1)) c= {0,1} by A8, XBOOLE_1:1; then consider r1 being XFinSequence of such that A12: len (p | ((2 * o) + 1)) = len r1 and A13: rng r1 c= {0,1} and A14: for i being Nat st i <= len (p | ((2 * o) + 1)) holds (Sum ((p | ((2 * o) + 1)) | i)) + (Sum (r1 | i)) = 1 * i and A15: for i being Nat st i in len (p | ((2 * o) + 1)) holds r1 . i = 1 - ((p | ((2 * o) + 1)) . i) by Th25; take R = r1 ^ r2; ::_thesis: ( R in W & S1[x,R] ) len p = (len r1) + (len r2) by A12, A11, AFINSQ_1:17; then A16: dom R = n by A7, AFINSQ_1:def_3; rng r2 c= rng p by A11, AFINSQ_1:25; then rng r2 c= {0,1} by A8, XBOOLE_1:1; then (rng r1) \/ (rng r2) c= {0,1} by A13, XBOOLE_1:8; then A17: rng R c= {0,1} by AFINSQ_1:26; ( (p | ((2 * o) + 1)) | (dom (p | ((2 * o) + 1))) = p | ((2 * o) + 1) & r1 | (dom r1) = r1 ) ; then A18: (Sum (p | ((2 * o) + 1))) + (Sum r1) = 1 * (len (p | ((2 * o) + 1))) by A12, A14; A19: 2 * o < (2 * o) + 1 by NAT_1:13; then 2 * o c= (2 * o) + 1 by NAT_1:39; then A20: (p | ((2 * o) + 1)) | (2 * o) = p | (2 * o) by RELAT_1:74; A21: (2 * o) + 1 c= dom p by A10, NAT_1:39; then A22: dom (p | ((2 * o) + 1)) = (2 * o) + 1 by RELAT_1:62; A23: len (p | ((2 * o) + 1)) = (2 * o) + 1 by A21, RELAT_1:62; then A24: p | ((2 * o) + 1) = ((p | ((2 * o) + 1)) | (2 * o)) ^ <%((p | ((2 * o) + 1)) . (2 * o))%> by AFINSQ_1:56; 2 * o in (2 * o) + 1 by A19, NAT_1:44; then (p | ((2 * o) + 1)) . (2 * o) = p . (2 * o) by A23, FUNCT_1:47; then Sum (p | ((2 * o) + 1)) = (Sum (p | (2 * o))) + (Sum <%(p . (2 * o))%>) by A24, A20, AFINSQ_2:55; then A25: Sum (p | ((2 * o) + 1)) = o + 1 by A10, AFINSQ_2:53; m + 1 = (Sum (p | ((2 * o) + 1))) + (Sum r2) by A9, A11, AFINSQ_2:55; then (Sum r1) + (Sum r2) = ((m + 1) - ((2 * o) + 1)) + (2 * o) by A18, A22, A25; then Sum (r1 ^ r2) = m by AFINSQ_2:55; hence R in W by A16, A17, CARD_FIN:40; ::_thesis: S1[x,R] thus S1[x,R] ::_thesis: verum proof let p9 be XFinSequence of ; ::_thesis: for k being Nat st x = p9 & (2 * k) + 1 = min* { N where N is Element of NAT : 2 * (Sum (p9 | N)) > N } holds ex r1, r2 being XFinSequence of st ( R = r1 ^ r2 & len r1 = (2 * k) + 1 & len r1 = len (p9 | ((2 * k) + 1)) & p9 = (p9 | ((2 * k) + 1)) ^ r2 & ( for o being Nat st o < (2 * k) + 1 holds r1 . o = 1 - (p9 . o) ) ) let k be Nat; ::_thesis: ( x = p9 & (2 * k) + 1 = min* { N where N is Element of NAT : 2 * (Sum (p9 | N)) > N } implies ex r1, r2 being XFinSequence of st ( R = r1 ^ r2 & len r1 = (2 * k) + 1 & len r1 = len (p9 | ((2 * k) + 1)) & p9 = (p9 | ((2 * k) + 1)) ^ r2 & ( for o being Nat st o < (2 * k) + 1 holds r1 . o = 1 - (p9 . o) ) ) ) assume A26: ( x = p9 & (2 * k) + 1 = min* { N where N is Element of NAT : 2 * (Sum (p9 | N)) > N } ) ; ::_thesis: ex r1, r2 being XFinSequence of st ( R = r1 ^ r2 & len r1 = (2 * k) + 1 & len r1 = len (p9 | ((2 * k) + 1)) & p9 = (p9 | ((2 * k) + 1)) ^ r2 & ( for o being Nat st o < (2 * k) + 1 holds r1 . o = 1 - (p9 . o) ) ) set q9 = p9 | ((2 * k) + 1); take r1 ; ::_thesis: ex r2 being XFinSequence of st ( R = r1 ^ r2 & len r1 = (2 * k) + 1 & len r1 = len (p9 | ((2 * k) + 1)) & p9 = (p9 | ((2 * k) + 1)) ^ r2 & ( for o being Nat st o < (2 * k) + 1 holds r1 . o = 1 - (p9 . o) ) ) take r2 ; ::_thesis: ( R = r1 ^ r2 & len r1 = (2 * k) + 1 & len r1 = len (p9 | ((2 * k) + 1)) & p9 = (p9 | ((2 * k) + 1)) ^ r2 & ( for o being Nat st o < (2 * k) + 1 holds r1 . o = 1 - (p9 . o) ) ) thus ( R = r1 ^ r2 & len r1 = (2 * k) + 1 & len r1 = len (p9 | ((2 * k) + 1)) & p9 = (p9 | ((2 * k) + 1)) ^ r2 ) by A6, A10, A12, A11, A21, A26, RELAT_1:62; ::_thesis: for o being Nat st o < (2 * k) + 1 holds r1 . o = 1 - (p9 . o) thus for i being Nat st i < (2 * k) + 1 holds r1 . i = 1 - (p9 . i) ::_thesis: verum proof let i be Nat; ::_thesis: ( i < (2 * k) + 1 implies r1 . i = 1 - (p9 . i) ) assume i < (2 * k) + 1 ; ::_thesis: r1 . i = 1 - (p9 . i) then A27: i in len (p | ((2 * o) + 1)) by A6, A10, A22, A26, NAT_1:44; then r1 . i = 1 - ((p | ((2 * o) + 1)) . i) by A15; hence r1 . i = 1 - (p9 . i) by A6, A26, A27, FUNCT_1:47; ::_thesis: verum end; end; end; consider F being Function of ((Choose (n,(m + 1),1,0)) \ (Domin_0 (n,(m + 1)))),W such that A28: for x being set st x in (Choose (n,(m + 1),1,0)) \ (Domin_0 (n,(m + 1))) holds S1[x,F . x] from FUNCT_2:sch_1(A4); W c= rng F proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in W or x in rng F ) assume x in W ; ::_thesis: x in rng F then consider p being XFinSequence of such that A29: p = x and A30: dom p = n and A31: rng p c= {0,1} and A32: Sum p = m by CARD_FIN:40; set M = { N where N is Element of NAT : 2 * (Sum (p | N)) < N } ; m < m + 1 by NAT_1:13; then 2 * m < 2 * (m + 1) by XREAL_1:68; then 2 * m < n by A1, XXREAL_0:2; then ( 2 * (Sum (p | n)) < n & n in NAT ) by A30, A32, ORDINAL1:def_12, RELAT_1:69; then A33: n in { N where N is Element of NAT : 2 * (Sum (p | N)) < N } ; { N where N is Element of NAT : 2 * (Sum (p | N)) < N } c= NAT proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in { N where N is Element of NAT : 2 * (Sum (p | N)) < N } or y in NAT ) assume y in { N where N is Element of NAT : 2 * (Sum (p | N)) < N } ; ::_thesis: y in NAT then ex i being Element of NAT st ( i = y & 2 * (Sum (p | i)) < i ) ; hence y in NAT ; ::_thesis: verum end; then reconsider M = { N where N is Element of NAT : 2 * (Sum (p | N)) < N } as non empty Subset of NAT by A33; ex k being Nat st ( (2 * k) + 1 = min* M & Sum (p | ((2 * k) + 1)) = k & (2 * k) + 1 <= dom p ) proof set mm = min* M; min* M in M by NAT_1:def_1; then A34: ex o being Element of NAT st ( min* M = o & 2 * (Sum (p | o)) < o ) ; then reconsider m1 = (min* M) - 1 as Element of NAT by NAT_1:20; A35: 2 * (Sum (p | (min* M))) < m1 + 1 by A34; A36: m1 < m1 + 1 by NAT_1:13; then m1 c= min* M by NAT_1:39; then A37: (p | (min* M)) | m1 = p | m1 by RELAT_1:74; min* M <= dom p by A30, A33, NAT_1:def_1; then A38: min* M c= dom p by NAT_1:39; then dom (p | (min* M)) = min* M by RELAT_1:62; then m1 in dom (p | (min* M)) by A36, NAT_1:44; then A39: (p | (min* M)) . m1 = p . m1 by FUNCT_1:47; m1 < m1 + 1 by NAT_1:13; then not m1 in M by NAT_1:def_1; then 2 * (Sum (p | m1)) >= m1 ; then A40: ( Sum <%(p . m1)%> = p . m1 & (2 * (Sum (p | m1))) + (2 * (p . m1)) >= m1 + 0 ) by AFINSQ_2:53, XREAL_1:7; reconsider S = Sum (p | (min* M)) as Element of NAT by ORDINAL1:def_12; take S ; ::_thesis: ( (2 * S) + 1 = min* M & Sum (p | ((2 * S) + 1)) = S & (2 * S) + 1 <= dom p ) A41: min* M <= dom p by A30, A33, NAT_1:def_1; len (p | (min* M)) = m1 + 1 by A38, RELAT_1:62; then p | (min* M) = ((p | (min* M)) | m1) ^ <%((p | (min* M)) . m1)%> by AFINSQ_1:56; then Sum (p | (min* M)) = (Sum (p | m1)) + (Sum <%(p . m1)%>) by A39, A37, AFINSQ_2:55; hence ( (2 * S) + 1 = min* M & Sum (p | ((2 * S) + 1)) = S & (2 * S) + 1 <= dom p ) by A41, A40, A35, NAT_1:9; ::_thesis: verum end; then consider k being Nat such that A42: (2 * k) + 1 = min* M and A43: Sum (p | ((2 * k) + 1)) = k and A44: (2 * k) + 1 <= dom p ; set k1 = (2 * k) + 1; consider q being XFinSequence of such that A45: p = (p | ((2 * k) + 1)) ^ q by Th1; rng (p | ((2 * k) + 1)) c= rng p by RELAT_1:70; then rng (p | ((2 * k) + 1)) c= {0,1} by A31, XBOOLE_1:1; then consider r being XFinSequence of such that A46: len r = len (p | ((2 * k) + 1)) and A47: rng r c= {0,1} and A48: for i being Nat st i <= len (p | ((2 * k) + 1)) holds (Sum ((p | ((2 * k) + 1)) | i)) + (Sum (r | i)) = 1 * i and A49: for i being Nat st i in len (p | ((2 * k) + 1)) holds r . i = 1 - ((p | ((2 * k) + 1)) . i) by Th25; set rq = r ^ q; A50: dom (r ^ q) = (len (p | ((2 * k) + 1))) + (len q) by A46, AFINSQ_1:def_3; A51: m = k + (Sum q) by A32, A43, A45, AFINSQ_2:55; dom (r ^ q) = (len (p | ((2 * k) + 1))) + (len q) by A46, AFINSQ_1:def_3; then A52: dom (r ^ q) = dom p by A45, AFINSQ_1:def_3; ( (p | ((2 * k) + 1)) | (dom (p | ((2 * k) + 1))) = p | ((2 * k) + 1) & r | (dom r) = r ) ; then A53: (Sum (p | ((2 * k) + 1))) + (Sum r) = 1 * (len (p | ((2 * k) + 1))) by A46, A48; rng q c= rng p by A45, AFINSQ_1:25; then rng q c= {0,1} by A31, XBOOLE_1:1; then (rng r) \/ (rng q) c= {0,1} by A47, XBOOLE_1:8; then A54: rng (r ^ q) c= {0,1} by AFINSQ_1:26; A55: (2 * k) + 1 c= dom p by A44, NAT_1:39; then A56: len (p | ((2 * k) + 1)) = (2 * k) + 1 by RELAT_1:62; then A57: (r ^ q) | ((2 * k) + 1) = r by A46, AFINSQ_1:57; A58: ((2 * k) + 1) + 1 > (2 * k) + 1 by NAT_1:13; then A59: (2 * k) + 1 < 2 * (Sum r) by A43, A53, A56; A60: 2 * (Sum r) > (2 * k) + 1 by A43, A53, A56, A58; then consider j being Nat such that A61: ( (2 * j) + 1 = min* { N where N is Element of NAT : 2 * (Sum ((r ^ q) | N)) > N } & (2 * j) + 1 <= dom (r ^ q) & j = Sum ((r ^ q) | (2 * j)) & (r ^ q) . (2 * j) = 1 ) by A54, A57, Th2, Th15; set j1 = (2 * j) + 1; A62: len ((p | ((2 * k) + 1)) ^ q) = (len (p | ((2 * k) + 1))) + (len q) by AFINSQ_1:def_3; not r ^ q is dominated_by_0 by A60, A57, Th2; then A63: not r ^ q in Domin_0 (n,(m + 1)) by Th20; set rqj = (r ^ q) | ((2 * j) + 1); Sum (r ^ q) = (Sum r) + (Sum q) by AFINSQ_2:55; then r ^ q in Choose (n,(m + 1),1,0) by A30, A43, A53, A56, A52, A54, A51, CARD_FIN:40; then A64: r ^ q in (Choose (n,(m + 1),1,0)) \ (Domin_0 (n,(m + 1))) by A63, XBOOLE_0:def_5; then consider r1, r2 being XFinSequence of such that A65: F . (r ^ q) = r1 ^ r2 and A66: len r1 = (2 * j) + 1 and A67: ( len r1 = len ((r ^ q) | ((2 * j) + 1)) & r ^ q = ((r ^ q) | ((2 * j) + 1)) ^ r2 ) and A68: for i being Nat st i < (2 * j) + 1 holds r1 . i = 1 - ((r ^ q) . i) by A28, A61; A69: dom (r ^ q) = (len r1) + (len r2) by A67, AFINSQ_1:def_3; then A70: len (r1 ^ r2) = len ((p | ((2 * k) + 1)) ^ q) by A50, A62, AFINSQ_1:17; reconsider K = { N where N is Element of NAT : 2 * (Sum ((r ^ q) | N)) > N } as non empty Subset of NAT by A61, NAT_1:def_1; (r ^ q) | ((2 * k) + 1) = r by A46, A56, AFINSQ_1:57; then (2 * k) + 1 in K by A59; then A71: (2 * k) + 1 >= (2 * j) + 1 by A61, NAT_1:def_1; then (2 * j) + 1 c= (2 * k) + 1 by NAT_1:39; then A72: (p | ((2 * k) + 1)) | ((2 * j) + 1) = p | ((2 * j) + 1) by RELAT_1:74; (2 * j) + 1 in K by A61, NAT_1:def_1; then A73: ex N being Element of NAT st ( N = (2 * j) + 1 & 2 * (Sum ((r ^ q) | N)) > N ) ; (Sum ((p | ((2 * k) + 1)) | ((2 * j) + 1))) + (Sum (r | ((2 * j) + 1))) = ((2 * j) + 1) * 1 by A48, A56, A71; then 2 * (Sum (r | ((2 * j) + 1))) = (2 * ((2 * j) + 1)) - (2 * (Sum (p | ((2 * j) + 1)))) by A72; then ((2 * j) + 1) + (((2 * j) + 1) - (2 * (Sum (p | ((2 * j) + 1))))) > (2 * (Sum (p | ((2 * j) + 1)))) + (((2 * j) + 1) - (2 * (Sum (p | ((2 * j) + 1))))) by A46, A56, A71, A73, AFINSQ_1:58; then (2 * j) + 1 > 2 * (Sum (p | ((2 * j) + 1))) by XREAL_1:6; then (2 * j) + 1 in M ; then (2 * j) + 1 >= (2 * k) + 1 by A42, NAT_1:def_1; then A74: (2 * j) + 1 = (2 * k) + 1 by A71, XXREAL_0:1; A75: len ((p | ((2 * k) + 1)) ^ q) = len (r ^ q) by A50, AFINSQ_1:def_3; now__::_thesis:_for_i_being_Nat_st_i_<_len_(r1_^_r2)_holds_ (r1_^_r2)_._i_=_((p_|_((2_*_k)_+_1))_^_q)_._i let i be Nat; ::_thesis: ( i < len (r1 ^ r2) implies (r1 ^ r2) . i = ((p | ((2 * k) + 1)) ^ q) . i ) assume A76: i < len (r1 ^ r2) ; ::_thesis: (r1 ^ r2) . i = ((p | ((2 * k) + 1)) ^ q) . i now__::_thesis:_(r1_^_r2)_._i_=_((p_|_((2_*_k)_+_1))_^_q)_._i percases ( i < len r1 or i >= len r1 ) ; supposeA77: i < len r1 ; ::_thesis: (r1 ^ r2) . i = ((p | ((2 * k) + 1)) ^ q) . i then A78: ( i in dom r1 & r1 . i = 1 - ((r ^ q) . i) ) by A66, A68, NAT_1:44; A79: i in len r1 by A77, NAT_1:44; A80: ( len r1 = len (p | ((2 * k) + 1)) & i in NAT ) by A55, A66, A74, ORDINAL1:def_12, RELAT_1:62; then A81: r . i = 1 - ((p | ((2 * k) + 1)) . i) by A49, A79; ( ((p | ((2 * k) + 1)) ^ q) . i = (p | ((2 * k) + 1)) . i & (r ^ q) . i = r . i ) by A46, A79, A80, AFINSQ_1:def_3; hence (r1 ^ r2) . i = ((p | ((2 * k) + 1)) ^ q) . i by A81, A78, AFINSQ_1:def_3; ::_thesis: verum end; supposeA82: i >= len r1 ; ::_thesis: (r1 ^ r2) . i = ((p | ((2 * k) + 1)) ^ q) . i then A83: ((p | ((2 * k) + 1)) ^ q) . i = q . (i - (len r)) by A46, A56, A66, A74, A70, A76, AFINSQ_1:19; ( (r1 ^ r2) . i = r2 . (i - (len r1)) & (r ^ q) . i = q . (i - (len r)) ) by A46, A56, A66, A74, A70, A75, A76, A82, AFINSQ_1:19; hence (r1 ^ r2) . i = ((p | ((2 * k) + 1)) ^ q) . i by A67, A70, A75, A76, A82, A83, AFINSQ_1:19; ::_thesis: verum end; end; end; hence (r1 ^ r2) . i = ((p | ((2 * k) + 1)) ^ q) . i ; ::_thesis: verum end; then A84: r1 ^ r2 = (p | ((2 * k) + 1)) ^ q by A69, A50, A62, AFINSQ_1:9, AFINSQ_1:17; r ^ q in dom F by A64, FUNCT_2:def_1; hence x in rng F by A29, A45, A65, A84, FUNCT_1:3; ::_thesis: verum end; then A85: rng F = W by XBOOLE_0:def_10; A86: F is one-to-one proof let x be set ; :: according to FUNCT_1:def_4 ::_thesis: for b1 being set holds ( not x in dom F or not b1 in dom F or not F . x = F . b1 or x = b1 ) let y be set ; ::_thesis: ( not x in dom F or not y in dom F or not F . x = F . y or x = y ) assume that A87: x in dom F and A88: y in dom F and A89: F . x = F . y ; ::_thesis: x = y x in Choose (n,(m + 1),1,0) by A87, XBOOLE_0:def_5; then consider p being XFinSequence of such that A90: p = x and A91: dom p = n and A92: rng p c= {0,1} and A93: Sum p = m + 1 by CARD_FIN:40; not p in Domin_0 (n,(m + 1)) by A87, A90, XBOOLE_0:def_5; then not p is dominated_by_0 by A91, A93, Def2; then consider k1 being Nat such that A94: ( (2 * k1) + 1 = min* { N where N is Element of NAT : 2 * (Sum (p | N)) > N } & (2 * k1) + 1 <= dom p & k1 = Sum (p | (2 * k1)) & p . (2 * k1) = 1 ) by A92, Th15; y in Choose (n,(m + 1),1,0) by A88, XBOOLE_0:def_5; then consider q being XFinSequence of such that A95: q = y and A96: dom q = n and A97: rng q c= {0,1} and A98: Sum q = m + 1 by CARD_FIN:40; not q in Domin_0 (n,(m + 1)) by A88, A95, XBOOLE_0:def_5; then not q is dominated_by_0 by A96, A98, Def2; then consider k2 being Nat such that A99: ( (2 * k2) + 1 = min* { N where N is Element of NAT : 2 * (Sum (q | N)) > N } & (2 * k2) + 1 <= dom q & k2 = Sum (q | (2 * k2)) & q . (2 * k2) = 1 ) by A97, Th15; A100: len q = n by A96; reconsider M = { N where N is Element of NAT : 2 * (Sum (q | N)) > N } as non empty Subset of NAT by A99, NAT_1:def_1; set qk = q | ((2 * k2) + 1); consider s1, s2 being XFinSequence of such that A101: F . y = s1 ^ s2 and A102: len s1 = (2 * k2) + 1 and A103: len s1 = len (q | ((2 * k2) + 1)) and A104: q = (q | ((2 * k2) + 1)) ^ s2 and A105: for i being Nat st i < (2 * k2) + 1 holds s1 . i = 1 - (q . i) by A28, A88, A95, A99; A106: len q = (len (q | ((2 * k2) + 1))) + (len s2) by A104, AFINSQ_1:17; then A107: len q = len (s1 ^ s2) by A103, AFINSQ_1:17; reconsider K = { N where N is Element of NAT : 2 * (Sum (p | N)) > N } as non empty Subset of NAT by A94, NAT_1:def_1; set pk = p | ((2 * k1) + 1); consider r1, r2 being XFinSequence of such that A108: F . x = r1 ^ r2 and A109: len r1 = (2 * k1) + 1 and A110: len r1 = len (p | ((2 * k1) + 1)) and A111: p = (p | ((2 * k1) + 1)) ^ r2 and A112: for i being Nat st i < (2 * k1) + 1 holds r1 . i = 1 - (p . i) by A28, A87, A90, A94; assume x <> y ; ::_thesis: contradiction then consider i being Nat such that A113: i < len p and A114: p . i <> q . i by A90, A91, A95, A96, A106, AFINSQ_1:9; A115: len p = (len (p | ((2 * k1) + 1))) + (len r2) by A111, AFINSQ_1:17; then A116: len p = len (r1 ^ r2) by A110, AFINSQ_1:17; now__::_thesis:_contradiction percases ( k1 = k2 or k1 > k2 or k1 < k2 ) by XXREAL_0:1; supposeA117: k1 = k2 ; ::_thesis: contradiction now__::_thesis:_contradiction percases ( i < len (p | ((2 * k1) + 1)) or i >= len (p | ((2 * k1) + 1)) ) ; supposeA118: i < len (p | ((2 * k1) + 1)) ; ::_thesis: contradiction then i in len (p | ((2 * k1) + 1)) by NAT_1:44; then A119: ( r1 . i = (r1 ^ r2) . i & s1 . i = (s1 ^ s2) . i ) by A109, A110, A102, A117, AFINSQ_1:def_3; ( r1 . i = 1 - (p . i) & s1 . i = 1 - (q . i) ) by A109, A110, A112, A105, A117, A118; hence contradiction by A89, A108, A101, A114, A119; ::_thesis: verum end; supposeA120: i >= len (p | ((2 * k1) + 1)) ; ::_thesis: contradiction then A121: (s1 ^ s2) . i = s2 . (i - (len (p | ((2 * k1) + 1)))) by A91, A109, A110, A96, A102, A107, A113, A117, AFINSQ_1:19; ( p . i = r2 . (i - (len (p | ((2 * k1) + 1)))) & q . i = s2 . (i - (len (p | ((2 * k1) + 1)))) ) by A91, A109, A110, A111, A102, A103, A104, A100, A113, A117, A120, AFINSQ_1:19; hence contradiction by A89, A108, A110, A101, A116, A113, A114, A120, A121, AFINSQ_1:19; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA122: k1 > k2 ; ::_thesis: contradiction len s1 <= len p by A91, A96, A103, A106, NAT_1:11; then A123: len s1 c= dom p by NAT_1:39; 2 * k2 < 2 * k1 by A122, XREAL_1:68; then A124: len s1 < len r1 by A109, A102, XREAL_1:6; then (s1 ^ s2) | (len s1) = r1 | (len s1) by A89, A108, A101, AFINSQ_1:58; then A125: s1 = r1 | (len s1) by AFINSQ_1:57; A126: for k being Nat st k < len (q | ((2 * k2) + 1)) holds (q | ((2 * k2) + 1)) . k = (p | (len (q | ((2 * k2) + 1)))) . k proof let k be Nat; ::_thesis: ( k < len (q | ((2 * k2) + 1)) implies (q | ((2 * k2) + 1)) . k = (p | (len (q | ((2 * k2) + 1)))) . k ) assume A127: k < len (q | ((2 * k2) + 1)) ; ::_thesis: (q | ((2 * k2) + 1)) . k = (p | (len (q | ((2 * k2) + 1)))) . k A128: k in len s1 by A103, A127, NAT_1:44; then A129: k in (dom q) /\ (len s1) by A91, A96, A123, XBOOLE_0:def_4; k in (dom p) /\ (len s1) by A123, A128, XBOOLE_0:def_4; then A130: p . k = (p | (len (q | ((2 * k2) + 1)))) . k by A103, FUNCT_1:48; A131: k < len r1 by A103, A124, A127, XXREAL_0:2; then A132: r1 . k = 1 - (p . k) by A109, A112; k in dom r1 by A131, NAT_1:44; then k in (dom r1) /\ (len s1) by A128, XBOOLE_0:def_4; then A133: r1 . k = (r1 | (len s1)) . k by FUNCT_1:48; s1 . k = 1 - (q . k) by A102, A103, A105, A127; hence (q | ((2 * k2) + 1)) . k = (p | (len (q | ((2 * k2) + 1)))) . k by A102, A125, A132, A133, A129, A130, FUNCT_1:48; ::_thesis: verum end; (2 * k2) + 1 in M by A99, NAT_1:def_1; then A134: ex N being Element of NAT st ( (2 * k2) + 1 = N & 2 * (Sum (q | N)) > N ) ; len (q | ((2 * k2) + 1)) = len (p | (len (q | ((2 * k2) + 1)))) by A103, A123, RELAT_1:62; then q | ((2 * k2) + 1) = p | (len (q | ((2 * k2) + 1))) by A126, AFINSQ_1:9; then len (q | ((2 * k2) + 1)) in K by A102, A103, A134; hence contradiction by A94, A109, A103, A124, NAT_1:def_1; ::_thesis: verum end; supposeA135: k1 < k2 ; ::_thesis: contradiction len r1 <= len q by A91, A110, A96, A115, NAT_1:11; then A136: len r1 c= dom q by NAT_1:39; 2 * k1 < 2 * k2 by A135, XREAL_1:68; then A137: len r1 < len s1 by A109, A102, XREAL_1:6; then (r1 ^ r2) | (len r1) = s1 | (len r1) by A89, A108, A101, AFINSQ_1:58; then A138: r1 = s1 | (len r1) by AFINSQ_1:57; A139: for k being Nat st k < len (p | ((2 * k1) + 1)) holds (p | ((2 * k1) + 1)) . k = (q | (len (p | ((2 * k1) + 1)))) . k proof let k be Nat; ::_thesis: ( k < len (p | ((2 * k1) + 1)) implies (p | ((2 * k1) + 1)) . k = (q | (len (p | ((2 * k1) + 1)))) . k ) assume A140: k < len (p | ((2 * k1) + 1)) ; ::_thesis: (p | ((2 * k1) + 1)) . k = (q | (len (p | ((2 * k1) + 1)))) . k A141: k in len r1 by A110, A140, NAT_1:44; then A142: k in (dom p) /\ (len r1) by A91, A96, A136, XBOOLE_0:def_4; k in (dom q) /\ (len r1) by A136, A141, XBOOLE_0:def_4; then A143: q . k = (q | (len (p | ((2 * k1) + 1)))) . k by A110, FUNCT_1:48; A144: k < len s1 by A110, A137, A140, XXREAL_0:2; then A145: s1 . k = 1 - (q . k) by A102, A105; k in dom s1 by A144, NAT_1:44; then k in (dom s1) /\ (len r1) by A141, XBOOLE_0:def_4; then A146: s1 . k = (s1 | (len r1)) . k by FUNCT_1:48; r1 . k = 1 - (p . k) by A109, A110, A112, A140; hence (p | ((2 * k1) + 1)) . k = (q | (len (p | ((2 * k1) + 1)))) . k by A109, A138, A145, A146, A142, A143, FUNCT_1:48; ::_thesis: verum end; (2 * k1) + 1 in K by A94, NAT_1:def_1; then A147: ex N being Element of NAT st ( (2 * k1) + 1 = N & 2 * (Sum (p | N)) > N ) ; len (p | ((2 * k1) + 1)) = len (q | (len (p | ((2 * k1) + 1)))) by A110, A136, RELAT_1:62; then p | ((2 * k1) + 1) = q | (len (p | ((2 * k1) + 1))) by A139, AFINSQ_1:9; then len (p | ((2 * k1) + 1)) in M by A109, A110, A147; hence contradiction by A110, A99, A102, A137, NAT_1:def_1; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; dom F = (Choose (n,(m + 1),1,0)) \ (Domin_0 (n,(m + 1))) by FUNCT_2:def_1; then (Choose (n,(m + 1),1,0)) \ (Domin_0 (n,(m + 1))),W are_equipotent by A86, A85, WELLORD2:def_4; hence card ((Choose (n,(m + 1),1,0)) \ (Domin_0 (n,(m + 1)))) = card (Choose (n,m,1,0)) by CARD_1:5; ::_thesis: verum end; theorem Th28: :: CATALAN2:28 for m, n being Nat st 2 * (m + 1) <= n holds card (Domin_0 (n,(m + 1))) = (n choose (m + 1)) - (n choose m) proof let m, n be Nat; ::_thesis: ( 2 * (m + 1) <= n implies card (Domin_0 (n,(m + 1))) = (n choose (m + 1)) - (n choose m) ) set CH = Choose (n,(m + 1),1,0); set Z = Domin_0 (n,(m + 1)); set W = Choose (n,m,1,0); A1: ( card (Choose (n,(m + 1),1,0)) = (card n) choose (m + 1) & card n = n ) by CARD_1:def_2, CARD_FIN:16; card ((Choose (n,(m + 1),1,0)) \ (Domin_0 (n,(m + 1)))) = (card (Choose (n,(m + 1),1,0))) - (card (Domin_0 (n,(m + 1)))) by Th21, CARD_2:44; then A2: card (Domin_0 (n,(m + 1))) = (card (Choose (n,(m + 1),1,0))) - (card ((Choose (n,(m + 1),1,0)) \ (Domin_0 (n,(m + 1))))) ; assume 2 * (m + 1) <= n ; ::_thesis: card (Domin_0 (n,(m + 1))) = (n choose (m + 1)) - (n choose m) then card (Domin_0 (n,(m + 1))) = (card (Choose (n,(m + 1),1,0))) - (card (Choose (n,m,1,0))) by A2, Th27; hence card (Domin_0 (n,(m + 1))) = (n choose (m + 1)) - (n choose m) by A1, CARD_FIN:16; ::_thesis: verum end; theorem Th29: :: CATALAN2:29 for m, n being Nat st 2 * m <= n holds card (Domin_0 (n,m)) = (((n + 1) - (2 * m)) / ((n + 1) - m)) * (n choose m) proof let m, n be Nat; ::_thesis: ( 2 * m <= n implies card (Domin_0 (n,m)) = (((n + 1) - (2 * m)) / ((n + 1) - m)) * (n choose m) ) assume A1: 2 * m <= n ; ::_thesis: card (Domin_0 (n,m)) = (((n + 1) - (2 * m)) / ((n + 1) - m)) * (n choose m) now__::_thesis:_card_(Domin_0_(n,m))_=_(((n_+_1)_-_(2_*_m))_/_((n_+_1)_-_m))_*_(n_choose_m) percases ( m = 0 or m > 0 ) ; supposeA2: m = 0 ; ::_thesis: card (Domin_0 (n,m)) = (((n + 1) - (2 * m)) / ((n + 1) - m)) * (n choose m) then n choose m = 1 by NEWTON:19; then (((n + 1) - (2 * m)) / ((n + 1) - m)) * (n choose m) = 1 by A2, XCMPLX_1:60; hence card (Domin_0 (n,m)) = (((n + 1) - (2 * m)) / ((n + 1) - m)) * (n choose m) by A2, Th24; ::_thesis: verum end; supposeA3: m > 0 ; ::_thesis: card (Domin_0 (n,m)) = (((n + 1) - (2 * m)) / ((n + 1) - m)) * (n choose m) A4: m <= m + m by NAT_1:11; then reconsider nm = n - m as Element of NAT by A1, NAT_1:21, XXREAL_0:2; reconsider m1 = m - 1 as Element of NAT by A3, NAT_1:20; set n9 = n ! ; set m9 = m ! ; set nm19 = (nm + 1) ! ; set nm9 = nm ! ; m <= n by A1, A4, XXREAL_0:2; then A5: n choose m = (n !) / ((m !) * (nm !)) by NEWTON:def_3; A6: 2 * (m1 + 1) <= n by A1; set m19 = m1 ! ; A7: 1 / ((m1 !) * ((nm + 1) !)) = ((m1 + 1) * 1) / (((m1 !) * ((nm + 1) !)) * (m1 + 1)) by XCMPLX_1:91 .= m / (((nm + 1) !) * ((m1 !) * (m1 + 1))) .= m / (((nm + 1) !) * ((m1 + 1) !)) by NEWTON:15 .= - ((- m) / (((nm + 1) !) * (m !))) by XCMPLX_1:190 ; 1 / ((m !) * (nm !)) = ((nm + 1) * 1) / (((m !) * (nm !)) * (nm + 1)) by XCMPLX_1:91 .= (nm + 1) / ((m !) * ((nm !) * (nm + 1))) .= (nm + 1) / ((m !) * ((nm + 1) !)) by NEWTON:15 ; then A8: (1 / ((m !) * (nm !))) - (1 / ((m1 !) * ((nm + 1) !))) = ((nm + 1) / ((m !) * ((nm + 1) !))) + ((- m) / ((m !) * ((nm + 1) !))) by A7 .= ((nm + 1) + (- m)) / ((m !) * ((nm + 1) !)) by XCMPLX_1:62 .= ((n + 1) - (2 * m)) / ((m !) * ((nm !) * (nm + 1))) by NEWTON:15 .= (1 * ((n + 1) - (2 * m))) / (((m !) * (nm !)) * (nm + 1)) .= (1 / ((m !) * (nm !))) * (((n + 1) - (2 * m)) / (nm + 1)) by XCMPLX_1:76 ; m1 <= m1 + ((1 + m1) + 1) by NAT_1:11; then A9: m1 <= n by A1, XXREAL_0:2; n - m1 = nm + 1 ; then A10: n choose m1 = (n !) / ((m1 !) * ((nm + 1) !)) by A9, NEWTON:def_3; ((n !) / ((m !) * (nm !))) - ((n !) / ((m1 !) * ((nm + 1) !))) = ((n !) * (1 / ((m !) * (nm !)))) - ((n !) / ((m1 !) * ((nm + 1) !))) by XCMPLX_1:99 .= ((n !) * (1 / ((m !) * (nm !)))) - ((n !) * (1 / ((m1 !) * ((nm + 1) !)))) by XCMPLX_1:99 .= (n !) * ((1 / ((m !) * (nm !))) - (1 / ((m1 !) * ((nm + 1) !)))) .= (n !) * ((1 / ((m !) * (nm !))) * (((n + 1) - (2 * m)) / (nm + 1))) by A8 .= ((n !) * (1 / ((m !) * (nm !)))) * (((n + 1) - (2 * m)) / (nm + 1)) .= (((n !) * 1) / ((m !) * (nm !))) * (((n + 1) - (2 * m)) / (nm + 1)) by XCMPLX_1:74 ; hence card (Domin_0 (n,m)) = (((n + 1) - (2 * m)) / ((n + 1) - m)) * (n choose m) by A5, A10, A6, Th28; ::_thesis: verum end; end; end; hence card (Domin_0 (n,m)) = (((n + 1) - (2 * m)) / ((n + 1) - m)) * (n choose m) ; ::_thesis: verum end; theorem Th30: :: CATALAN2:30 for k being Nat holds card (Domin_0 ((2 + k),1)) = k + 1 proof let k be Nat; ::_thesis: card (Domin_0 ((2 + k),1)) = k + 1 card (Domin_0 ((2 + k),1)) = ((((2 + k) + 1) - (2 * 1)) / (((2 + k) + 1) - 1)) * ((2 + k) choose 1) by Th29, NAT_1:11 .= ((k + 1) / (2 + k)) * (2 + k) by STIRL2_1:51 .= k + 1 by XCMPLX_1:87 ; hence card (Domin_0 ((2 + k),1)) = k + 1 ; ::_thesis: verum end; theorem :: CATALAN2:31 for k being Nat holds card (Domin_0 ((4 + k),2)) = ((k + 1) * (k + 4)) / 2 proof let k be Nat; ::_thesis: card (Domin_0 ((4 + k),2)) = ((k + 1) * (k + 4)) / 2 card (Domin_0 ((4 + k),2)) = ((((4 + k) + 1) - (2 * 2)) / (((4 + k) + 1) - 2)) * ((4 + k) choose 2) by Th29, NAT_1:11 .= ((k + 1) / (k + 3)) * (((4 + k) * ((4 + k) - 1)) / 2) by STIRL2_1:51 .= ((((k + 1) / (k + 3)) * (3 + k)) * (4 + k)) / 2 .= ((k + 1) * (4 + k)) / 2 by XCMPLX_1:87 ; hence card (Domin_0 ((4 + k),2)) = ((k + 1) * (k + 4)) / 2 ; ::_thesis: verum end; theorem :: CATALAN2:32 for k being Nat holds card (Domin_0 ((6 + k),3)) = (((k + 1) * (k + 5)) * (k + 6)) / 6 proof let k be Nat; ::_thesis: card (Domin_0 ((6 + k),3)) = (((k + 1) * (k + 5)) * (k + 6)) / 6 card (Domin_0 ((6 + k),3)) = ((((6 + k) + 1) - (2 * 3)) / (((6 + k) + 1) - 3)) * ((6 + k) choose 3) by Th29, NAT_1:11 .= ((k + 1) / (k + 4)) * ((((6 + k) * ((6 + k) - 1)) * ((6 + k) - 2)) / 6) by STIRL2_1:51 .= (((((k + 1) / (k + 4)) * (4 + k)) * (5 + k)) * (6 + k)) / 6 .= (((k + 1) * (5 + k)) * (6 + k)) / 6 by XCMPLX_1:87 ; hence card (Domin_0 ((6 + k),3)) = (((k + 1) * (k + 5)) * (k + 6)) / 6 ; ::_thesis: verum end; theorem Th33: :: CATALAN2:33 for n being Nat holds card (Domin_0 ((2 * n),n)) = ((2 * n) choose n) / (n + 1) proof let n be Nat; ::_thesis: card (Domin_0 ((2 * n),n)) = ((2 * n) choose n) / (n + 1) card (Domin_0 ((2 * n),n)) = ((((2 * n) + 1) - (2 * n)) / (((2 * n) + 1) - n)) * ((2 * n) choose n) by Th29 .= (1 * ((2 * n) choose n)) / (n + 1) by XCMPLX_1:74 ; hence card (Domin_0 ((2 * n),n)) = ((2 * n) choose n) / (n + 1) ; ::_thesis: verum end; theorem Th34: :: CATALAN2:34 for n being Nat holds card (Domin_0 ((2 * n),n)) = Catalan (n + 1) proof let n be Nat; ::_thesis: card (Domin_0 ((2 * n),n)) = Catalan (n + 1) A1: Catalan (n + 1) = (((2 * (n + 1)) -' 2) choose ((n + 1) -' 1)) / (n + 1) by CATALAN1:def_1; ( ((2 * n) + 2) -' 2 = ((2 * n) + 2) - 2 & (n + 1) -' 1 = (n + 1) - 1 ) by XREAL_0:def_2; hence card (Domin_0 ((2 * n),n)) = Catalan (n + 1) by A1, Th33; ::_thesis: verum end; definition let D be set ; mode OMEGA of D -> non empty functional set means :Def3: :: CATALAN2:def 3 for x being set st x in it holds x is XFinSequence of ; existence ex b1 being non empty functional set st for x being set st x in b1 holds x is XFinSequence of proof reconsider D9OMEGA = D ^omega as non empty functional set ; take D9OMEGA ; ::_thesis: for x being set st x in D9OMEGA holds x is XFinSequence of thus for x being set st x in D9OMEGA holds x is XFinSequence of by AFINSQ_1:def_7; ::_thesis: verum end; end; :: deftheorem Def3 defines OMEGA CATALAN2:def_3_:_ for D being set for b2 being non empty functional set holds ( b2 is OMEGA of D iff for x being set st x in b2 holds x is XFinSequence of ); definition let D be set ; :: original: ^omega redefine funcD ^omega -> OMEGA of D; coherence D ^omega is OMEGA of D proof ( D ^omega is functional & ( for x being set st x in D ^omega holds x is XFinSequence of ) ) by AFINSQ_1:def_7; hence D ^omega is OMEGA of D by Def3; ::_thesis: verum end; end; registration let D be set ; let F be OMEGA of D; cluster -> T-Sequence-like D -valued finite for Element of F; coherence for b1 being Element of F holds ( b1 is finite & b1 is D -valued & b1 is T-Sequence-like ) by Def3; end; theorem :: CATALAN2:35 for n being Nat holds card { pN where pN is Element of NAT ^omega : ( dom pN = 2 * n & pN is dominated_by_0 ) } = (2 * n) choose n proof let n be Nat; ::_thesis: card { pN where pN is Element of NAT ^omega : ( dom pN = 2 * n & pN is dominated_by_0 ) } = (2 * n) choose n set D = bool ({0,1} ^omega); set 2n = 2 * n; defpred S1[ set , set ] means for i being Nat st i = $1 holds $2 = Domin_0 ((2 * n),i); set Z = { pN where pN is Element of NAT ^omega : ( dom pN = 2 * n & pN is dominated_by_0 ) } ; A1: for k being Nat st k in n + 1 holds ex x being Element of bool ({0,1} ^omega) st S1[k,x] proof let k be Nat; ::_thesis: ( k in n + 1 implies ex x being Element of bool ({0,1} ^omega) st S1[k,x] ) assume k in n + 1 ; ::_thesis: ex x being Element of bool ({0,1} ^omega) st S1[k,x] reconsider Z = Domin_0 ((2 * n),k) as Element of bool ({0,1} ^omega) ; take Z ; ::_thesis: S1[k,Z] thus S1[k,Z] ; ::_thesis: verum end; consider r being XFinSequence of such that A2: ( dom r = n + 1 & ( for k being Nat st k in n + 1 holds S1[k,r . k] ) ) from STIRL2_1:sch_5(A1); A3: { pN where pN is Element of NAT ^omega : ( dom pN = 2 * n & pN is dominated_by_0 ) } c= union (rng r) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { pN where pN is Element of NAT ^omega : ( dom pN = 2 * n & pN is dominated_by_0 ) } or x in union (rng r) ) assume x in { pN where pN is Element of NAT ^omega : ( dom pN = 2 * n & pN is dominated_by_0 ) } ; ::_thesis: x in union (rng r) then consider pN being Element of NAT ^omega such that A4: x = pN and A5: ( dom pN = 2 * n & pN is dominated_by_0 ) ; pN in Domin_0 ((2 * n),(Sum pN)) by A5, Th20; then 2 * (Sum pN) <= 2 * n by Th22; then (1 / 2) * (2 * (Sum pN)) <= (1 / 2) * (2 * n) by XREAL_1:64; then Sum pN < n + 1 by NAT_1:13; then A6: Sum pN in n + 1 by NAT_1:44; then r . (Sum pN) = Domin_0 ((2 * n),(Sum pN)) by A2; then A7: pN in r . (Sum pN) by A5, Th20; r . (Sum pN) in rng r by A2, A6, FUNCT_1:3; hence x in union (rng r) by A4, A7, TARSKI:def_4; ::_thesis: verum end; A8: union (rng r) c= { pN where pN is Element of NAT ^omega : ( dom pN = 2 * n & pN is dominated_by_0 ) } proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union (rng r) or x in { pN where pN is Element of NAT ^omega : ( dom pN = 2 * n & pN is dominated_by_0 ) } ) assume x in union (rng r) ; ::_thesis: x in { pN where pN is Element of NAT ^omega : ( dom pN = 2 * n & pN is dominated_by_0 ) } then consider y being set such that A9: x in y and A10: y in rng r by TARSKI:def_4; consider i being set such that A11: i in dom r and A12: y = r . i by A10, FUNCT_1:def_3; reconsider i = i as Element of NAT by A11; y = Domin_0 ((2 * n),i) by A2, A11, A12; then consider p being XFinSequence of such that A13: ( p = x & p is dominated_by_0 & dom p = 2 * n ) and Sum p = i by A9, Def2; p is Element of NAT ^omega by AFINSQ_1:def_7; hence x in { pN where pN is Element of NAT ^omega : ( dom pN = 2 * n & pN is dominated_by_0 ) } by A13; ::_thesis: verum end; A14: for i, j being Nat st i in dom r & j in dom r & i <> j holds r . i misses r . j proof let i, j be Nat; ::_thesis: ( i in dom r & j in dom r & i <> j implies r . i misses r . j ) assume that A15: i in dom r and A16: j in dom r and A17: i <> j ; ::_thesis: r . i misses r . j assume r . i meets r . j ; ::_thesis: contradiction then (r . i) /\ (r . j) <> {} by XBOOLE_0:def_7; then consider x being set such that A18: x in (r . i) /\ (r . j) by XBOOLE_0:def_1; A19: x in r . j by A18, XBOOLE_0:def_4; r . j = Domin_0 ((2 * n),j) by A2, A16; then A20: ex q being XFinSequence of st ( q = x & q is dominated_by_0 & dom q = 2 * n & Sum q = j ) by A19, Def2; A21: x in r . i by A18, XBOOLE_0:def_4; r . i = Domin_0 ((2 * n),i) by A2, A15; then ex p being XFinSequence of st ( p = x & p is dominated_by_0 & dom p = 2 * n & Sum p = i ) by A21, Def2; hence contradiction by A17, A20; ::_thesis: verum end; A22: for i being Nat st i in dom r holds r . i is finite proof let i be Nat; ::_thesis: ( i in dom r implies r . i is finite ) assume i in dom r ; ::_thesis: r . i is finite then r . i = Domin_0 ((2 * n),i) by A2; hence r . i is finite ; ::_thesis: verum end; consider Cardr being XFinSequence of such that A23: dom Cardr = dom r and A24: for i being Nat st i in dom Cardr holds Cardr . i = card (r . i) and A25: card (union (rng r)) = Sum Cardr by A22, A14, STIRL2_1:66; A26: ( n < dom Cardr & Cardr | (n + 1) = Cardr ) by A2, A23, NAT_1:13, RELAT_1:69; defpred S2[ Nat] means ( $1 < dom Cardr implies Sum (Cardr | ($1 + 1)) = (2 * n) choose $1 ); A27: S2[ 0 ] proof 0 in n + 1 by NAT_1:44; then r . 0 = Domin_0 ((2 * n),0) by A2; then A28: card (r . 0) = 1 by Th24; A29: 0 in 1 by NAT_1:44; assume A30: 0 < dom Cardr ; ::_thesis: Sum (Cardr | (0 + 1)) = (2 * n) choose 0 then 1 <= len Cardr by NAT_1:14; then A31: 1 c= dom Cardr by NAT_1:39; then A32: len (Cardr | 1) = 1 by RELAT_1:62; dom (Cardr | 1) = 1 by A31, RELAT_1:62; then (Cardr | 1) . 0 = Cardr . 0 by A29, FUNCT_1:47; then A33: Cardr | 1 = <%(Cardr . 0)%> by A32, AFINSQ_1:34; 0 in len Cardr by A30, NAT_1:44; then Cardr . 0 = card (r . 0) by A24; then Sum (Cardr | 1) = 1 by A33, A28, AFINSQ_2:53; hence Sum (Cardr | (0 + 1)) = (2 * n) choose 0 by NEWTON:19; ::_thesis: verum end; A34: for i being Nat st S2[i] holds S2[i + 1] proof let i be Nat; ::_thesis: ( S2[i] implies S2[i + 1] ) assume A35: S2[i] ; ::_thesis: S2[i + 1] set i1 = i + 1; assume A36: i + 1 < dom Cardr ; ::_thesis: Sum (Cardr | ((i + 1) + 1)) = (2 * n) choose (i + 1) then A37: i + 1 in dom Cardr by NAT_1:44; then A38: ( (Sum (Cardr | (i + 1))) + (Cardr . (i + 1)) = Sum (Cardr | ((i + 1) + 1)) & Cardr . (i + 1) = card (r . (i + 1)) ) by A24, AFINSQ_2:65; i + 1 <= n by A2, A23, A36, NAT_1:13; then A39: 2 * (i + 1) <= 2 * n by XREAL_1:64; r . (i + 1) = Domin_0 ((2 * n),(i + 1)) by A2, A23, A37; then Sum (Cardr | ((i + 1) + 1)) = ((2 * n) choose i) + (((2 * n) choose (i + 1)) - ((2 * n) choose i)) by A35, A36, A38, A39, Th28, NAT_1:13; hence Sum (Cardr | ((i + 1) + 1)) = (2 * n) choose (i + 1) ; ::_thesis: verum end; for i being Nat holds S2[i] from NAT_1:sch_2(A27, A34); then Sum Cardr = (2 * n) choose n by A26; hence card { pN where pN is Element of NAT ^omega : ( dom pN = 2 * n & pN is dominated_by_0 ) } = (2 * n) choose n by A25, A3, A8, XBOOLE_0:def_10; ::_thesis: verum end; theorem Th36: :: CATALAN2:36 for n, m, k, j, l being Nat st j = n - (2 * (k + 1)) & l = m - (k + 1) holds card { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } = (card (Domin_0 ((2 * k),k))) * (card (Domin_0 (j,l))) proof set q1 = 1 --> 1; set q0 = 1 --> 0; let n, m, k, j, l be Nat; ::_thesis: ( j = n - (2 * (k + 1)) & l = m - (k + 1) implies card { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } = (card (Domin_0 ((2 * k),k))) * (card (Domin_0 (j,l))) ) assume A1: ( j = n - (2 * (k + 1)) & l = m - (k + 1) ) ; ::_thesis: card { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } = (card (Domin_0 ((2 * k),k))) * (card (Domin_0 (j,l))) defpred S1[ set , set ] means ex r1, r2 being XFinSequence of st ( $1 = (((1 --> 0) ^ r1) ^ (1 --> 1)) ^ r2 & len (((1 --> 0) ^ r1) ^ (1 --> 1)) = 2 * (k + 1) & $2 = [r1,r2] ); set Z2 = Domin_0 (j,l); set Z1 = Domin_0 ((2 * k),k); set F = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } ; set 2k1 = 2 * (k + 1); A2: for x being set st x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } holds ex y being set st ( y in [:(Domin_0 ((2 * k),k)),(Domin_0 (j,l)):] & S1[x,y] ) proof A3: ( dom (1 --> 0) = 1 & Sum (1 --> 0) = 0 * 1 ) by AFINSQ_2:58, FUNCOP_1:13; let x be set ; ::_thesis: ( x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } implies ex y being set st ( y in [:(Domin_0 ((2 * k),k)),(Domin_0 (j,l)):] & S1[x,y] ) ) assume x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } ; ::_thesis: ex y being set st ( y in [:(Domin_0 ((2 * k),k)),(Domin_0 (j,l)):] & S1[x,y] ) then consider pN being Element of NAT ^omega such that A4: ( pN = x & pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) ; 2 * (k + 1) > 2 * 0 by XREAL_1:68; then reconsider M = { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } as non empty Subset of NAT by A4, NAT_1:def_1; consider r2 being XFinSequence of such that A5: pN = (pN | (2 * (k + 1))) ^ r2 by Th1; ( 2 * (k + 1) > 2 * 0 & pN is dominated_by_0 ) by A4, Th20, XREAL_1:68; then consider r1 being XFinSequence of such that A6: pN | (2 * (k + 1)) = ((1 --> 0) ^ r1) ^ (1 --> 1) and A7: r1 is dominated_by_0 by A4, Th14; A8: Sum (1 --> 1) = 1 * 1 by AFINSQ_2:58; 2 * (k + 1) in M by A4, NAT_1:def_1; then A9: ex o being Element of NAT st ( o = 2 * (k + 1) & 2 * (Sum (pN | o)) = o & o > 0 ) ; then k + 1 = (Sum ((1 --> 0) ^ r1)) + (Sum (1 --> 1)) by A6, AFINSQ_2:55; then A10: k = (Sum (1 --> 0)) + (Sum r1) by A8, AFINSQ_2:55; pN is dominated_by_0 by A4, Th20; then A11: r2 is dominated_by_0 by A5, A9, Th12; pN is dominated_by_0 by A4, Th20; then A12: len (pN | (2 * (k + 1))) = 2 * (k + 1) by A9, Th11; Sum pN = m by A4, Th20; then A13: m = (k + 1) + (Sum r2) by A5, A9, AFINSQ_2:55; take [r1,r2] ; ::_thesis: ( [r1,r2] in [:(Domin_0 ((2 * k),k)),(Domin_0 (j,l)):] & S1[x,[r1,r2]] ) A14: dom (1 --> 1) = 1 by FUNCOP_1:13; dom pN = n by A4, Th20; then n = (2 * (k + 1)) + (len r2) by A5, A12, AFINSQ_1:def_3; then A15: r2 in Domin_0 (j,l) by A1, A13, A11, Th20; 2 * (k + 1) = (len ((1 --> 0) ^ r1)) + (len (1 --> 1)) by A6, A12, AFINSQ_1:17; then (2 * k) + 1 = (len (1 --> 0)) + (len r1) by A14, AFINSQ_1:17; then r1 in Domin_0 ((2 * k),k) by A7, A10, A3, Th20; hence ( [r1,r2] in [:(Domin_0 ((2 * k),k)),(Domin_0 (j,l)):] & S1[x,[r1,r2]] ) by A4, A5, A6, A12, A15, ZFMISC_1:def_2; ::_thesis: verum end; consider f being Function of { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } ,[:(Domin_0 ((2 * k),k)),(Domin_0 (j,l)):] such that A16: for x being set st x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } holds S1[x,f . x] from FUNCT_2:sch_1(A2); A17: ( [:(Domin_0 ((2 * k),k)),(Domin_0 (j,l)):] = {} implies { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } = {} ) proof assume [:(Domin_0 ((2 * k),k)),(Domin_0 (j,l)):] = {} ; ::_thesis: { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } = {} then ( Domin_0 ((2 * k),k) = {} or Domin_0 (j,l) = {} ) ; then 2 * l > j by Th22; then (2 * m) - (2 * (k + 1)) > n - (2 * (k + 1)) by A1; then A18: 2 * m > n by XREAL_1:9; assume { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } <> {} ; ::_thesis: contradiction then consider x being set such that A19: x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } by XBOOLE_0:def_1; ex pN being Element of NAT ^omega st ( pN = x & pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) by A19; hence contradiction by A18, Th22; ::_thesis: verum end; then A20: dom f = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } by FUNCT_2:def_1; A21: rng f = [:(Domin_0 ((2 * k),k)),(Domin_0 (j,l)):] proof A22: ( Sum (1 --> 0) = 1 * 0 & Sum (1 --> 1) = 1 * 1 ) by AFINSQ_2:58; thus rng f c= [:(Domin_0 ((2 * k),k)),(Domin_0 (j,l)):] ; :: according to XBOOLE_0:def_10 ::_thesis: [:(Domin_0 ((2 * k),k)),(Domin_0 (j,l)):] c= rng f A23: dom (1 --> 0) = 1 by FUNCOP_1:13; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in [:(Domin_0 ((2 * k),k)),(Domin_0 (j,l)):] or x in rng f ) assume x in [:(Domin_0 ((2 * k),k)),(Domin_0 (j,l)):] ; ::_thesis: x in rng f then consider x1, x2 being set such that A24: x1 in Domin_0 ((2 * k),k) and A25: x2 in Domin_0 (j,l) and A26: x = [x1,x2] by ZFMISC_1:def_2; consider p being XFinSequence of such that A27: p = x1 and A28: p is dominated_by_0 and A29: dom p = 2 * k and A30: Sum p = k by A24, Def2; consider q being XFinSequence of such that A31: q = x2 and A32: q is dominated_by_0 and A33: dom q = j and A34: Sum q = l by A25, Def2; set 0p1 = ((1 --> 0) ^ p) ^ (1 --> 1); A35: dom ((((1 --> 0) ^ p) ^ (1 --> 1)) ^ q) = (len (((1 --> 0) ^ p) ^ (1 --> 1))) + (len q) by AFINSQ_1:def_3; ( dom (((1 --> 0) ^ p) ^ (1 --> 1)) = (len ((1 --> 0) ^ p)) + (len (1 --> 1)) & dom (1 --> 1) = 1 ) by AFINSQ_1:def_3, FUNCOP_1:13; then A36: dom (((1 --> 0) ^ p) ^ (1 --> 1)) = ((len (1 --> 0)) + (len p)) + 1 by AFINSQ_1:17; then ((((1 --> 0) ^ p) ^ (1 --> 1)) ^ q) | ((2 * 1) + (len p)) = ((1 --> 0) ^ p) ^ (1 --> 1) by A23, AFINSQ_1:57; then A37: min* { N where N is Element of NAT : ( 2 * (Sum (((((1 --> 0) ^ p) ^ (1 --> 1)) ^ q) | N)) = N & N > 0 ) } = (2 * 1) + (len p) by A28, A29, A30, Th16; 1 <= (1 + (len p)) - (2 * (Sum p)) by A29, A30; then ((1 --> 0) ^ p) ^ (1 --> 1) is dominated_by_0 by A28, Th10; then A38: (((1 --> 0) ^ p) ^ (1 --> 1)) ^ q is dominated_by_0 by A32, Th7; A39: (((1 --> 0) ^ p) ^ (1 --> 1)) ^ q is Element of NAT ^omega by AFINSQ_1:def_7; ((1 --> 0) ^ p) ^ (1 --> 1) = (1 --> 0) ^ (p ^ (1 --> 1)) by AFINSQ_1:27; then Sum (((1 --> 0) ^ p) ^ (1 --> 1)) = (Sum (1 --> 0)) + (Sum (p ^ (1 --> 1))) by AFINSQ_2:55; then Sum (((1 --> 0) ^ p) ^ (1 --> 1)) = 0 + ((Sum p) + 1) by A22, AFINSQ_2:55; then Sum ((((1 --> 0) ^ p) ^ (1 --> 1)) ^ q) = (k + 1) + l by A30, A34, AFINSQ_2:55; then (((1 --> 0) ^ p) ^ (1 --> 1)) ^ q in Domin_0 (n,m) by A1, A29, A33, A38, A36, A23, A35, Th20; then A40: (((1 --> 0) ^ p) ^ (1 --> 1)) ^ q in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } by A29, A37, A39; then consider r1, r2 being XFinSequence of such that A41: (((1 --> 0) ^ p) ^ (1 --> 1)) ^ q = (((1 --> 0) ^ r1) ^ (1 --> 1)) ^ r2 and A42: len (((1 --> 0) ^ r1) ^ (1 --> 1)) = 2 * (k + 1) and A43: f . ((((1 --> 0) ^ p) ^ (1 --> 1)) ^ q) = [r1,r2] by A16; A44: ((((1 --> 0) ^ p) ^ (1 --> 1)) ^ q) | (2 * (k + 1)) = ((1 --> 0) ^ p) ^ (1 --> 1) by A29, A36, A23, AFINSQ_1:57; then (1 --> 0) ^ p = (1 --> 0) ^ r1 by A41, A42, AFINSQ_1:28, AFINSQ_1:57; then A45: p = r1 by AFINSQ_1:28; ((((1 --> 0) ^ r1) ^ (1 --> 1)) ^ r2) | (2 * (k + 1)) = ((1 --> 0) ^ r1) ^ (1 --> 1) by A42, AFINSQ_1:57; then q = r2 by A41, A44, AFINSQ_1:28; hence x in rng f by A20, A26, A27, A31, A40, A43, A45, FUNCT_1:3; ::_thesis: verum end; for x, y being set st x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } & y in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } & f . x = f . y holds x = y proof let x, y be set ; ::_thesis: ( x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } & y in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } & f . x = f . y implies x = y ) assume that A46: x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } and A47: y in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } and A48: f . x = f . y ; ::_thesis: x = y consider y1, y2 being XFinSequence of such that A49: y = (((1 --> 0) ^ y1) ^ (1 --> 1)) ^ y2 and len (((1 --> 0) ^ y1) ^ (1 --> 1)) = 2 * (k + 1) and A50: f . y = [y1,y2] by A16, A47; consider x1, x2 being XFinSequence of such that A51: x = (((1 --> 0) ^ x1) ^ (1 --> 1)) ^ x2 and len (((1 --> 0) ^ x1) ^ (1 --> 1)) = 2 * (k + 1) and A52: f . x = [x1,x2] by A16, A46; x1 = y1 by A48, A52, A50, XTUPLE_0:1; hence x = y by A48, A51, A52, A49, A50, XTUPLE_0:1; ::_thesis: verum end; then f is one-to-one by A17, FUNCT_2:19; then { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } ,[:(Domin_0 ((2 * k),k)),(Domin_0 (j,l)):] are_equipotent by A20, A21, WELLORD2:def_4; then card { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } = card [:(Domin_0 ((2 * k),k)),(Domin_0 (j,l)):] by CARD_1:5; hence card { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } = (card (Domin_0 ((2 * k),k))) * (card (Domin_0 (j,l))) by CARD_2:46; ::_thesis: verum end; theorem Th37: :: CATALAN2:37 for n, m being Nat st 2 * m <= n holds ex CardF being XFinSequence of st ( card { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } = Sum CardF & dom CardF = m & ( for j being Nat st j < m holds CardF . j = (card (Domin_0 ((2 * j),j))) * (card (Domin_0 ((n -' (2 * (j + 1))),(m -' (j + 1))))) ) ) proof let n, m be Nat; ::_thesis: ( 2 * m <= n implies ex CardF being XFinSequence of st ( card { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } = Sum CardF & dom CardF = m & ( for j being Nat st j < m holds CardF . j = (card (Domin_0 ((2 * j),j))) * (card (Domin_0 ((n -' (2 * (j + 1))),(m -' (j + 1))))) ) ) ) assume A1: 2 * m <= n ; ::_thesis: ex CardF being XFinSequence of st ( card { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } = Sum CardF & dom CardF = m & ( for j being Nat st j < m holds CardF . j = (card (Domin_0 ((2 * j),j))) * (card (Domin_0 ((n -' (2 * (j + 1))),(m -' (j + 1))))) ) ) set Z = Domin_0 (n,m); defpred S1[ set , set ] means for j being Nat st j = $1 holds $2 = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (j + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } ; set W = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } ; A2: for k being Nat st k in m holds ex x being Element of bool (Domin_0 (n,m)) st S1[k,x] proof let k be Nat; ::_thesis: ( k in m implies ex x being Element of bool (Domin_0 (n,m)) st S1[k,x] ) assume k in m ; ::_thesis: ex x being Element of bool (Domin_0 (n,m)) st S1[k,x] set NN = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } ; { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } c= Domin_0 (n,m) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } or x in Domin_0 (n,m) ) assume x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } ; ::_thesis: x in Domin_0 (n,m) then ex pN being Element of NAT ^omega st ( x = pN & pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) ; hence x in Domin_0 (n,m) ; ::_thesis: verum end; then reconsider NN = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (k + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } as Element of bool (Domin_0 (n,m)) ; take NN ; ::_thesis: S1[k,NN] thus S1[k,NN] ; ::_thesis: verum end; consider C being XFinSequence of such that A3: ( dom C = m & ( for k being Nat st k in m holds S1[k,C . k] ) ) from STIRL2_1:sch_5(A2); A4: { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } c= union (rng C) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } or x in union (rng C) ) assume x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } ; ::_thesis: x in union (rng C) then consider pN being Element of NAT ^omega such that A5: ( x = pN & pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) ; set I = { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ; { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } c= NAT proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } or y in NAT ) assume y in { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ; ::_thesis: y in NAT then ex i being Element of NAT st ( i = y & 2 * (Sum (pN | i)) = i & i > 0 ) ; hence y in NAT ; ::_thesis: verum end; then reconsider I = { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } as non empty Subset of NAT by A5; min* I in I by NAT_1:def_1; then consider M being Element of NAT such that A6: min* I = M and A7: 2 * (Sum (pN | M)) = M and A8: M > 0 ; Sum (pN | M) > 0 by A7, A8; then reconsider Sum1 = (Sum (pN | M)) - 1 as Nat by NAT_1:20; consider q being XFinSequence of such that A9: pN = (pN | M) ^ q by Th1; Sum pN = (Sum (pN | M)) + (Sum q) by A9, AFINSQ_2:55; then m = (Sum (pN | M)) + (Sum q) by A5, Th20; then A10: m >= Sum (pN | M) by NAT_1:11; Sum1 + 1 > Sum1 by NAT_1:13; then m > Sum1 by A10, XXREAL_0:2; then A11: Sum1 in m by NAT_1:44; then C . Sum1 = { qN where qN is Element of NAT ^omega : ( qN in Domin_0 (n,m) & 2 * (Sum1 + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (qN | N)) = N & N > 0 ) } ) } by A3; then A12: pN in C . Sum1 by A5, A6, A7; C . Sum1 in rng C by A3, A11, FUNCT_1:3; hence x in union (rng C) by A5, A12, TARSKI:def_4; ::_thesis: verum end; A13: for i, j being Nat st i in dom C & j in dom C & i <> j holds C . i misses C . j proof let i, j be Nat; ::_thesis: ( i in dom C & j in dom C & i <> j implies C . i misses C . j ) assume that A14: i in dom C and A15: j in dom C and A16: i <> j ; ::_thesis: C . i misses C . j assume C . i meets C . j ; ::_thesis: contradiction then (C . i) /\ (C . j) <> {} by XBOOLE_0:def_7; then consider x being set such that A17: x in (C . i) /\ (C . j) by XBOOLE_0:def_1; A18: x in C . j by A17, XBOOLE_0:def_4; C . j = { qN where qN is Element of NAT ^omega : ( qN in Domin_0 (n,m) & 2 * (j + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (qN | N)) = N & N > 0 ) } ) } by A3, A15; then A19: ex qN being Element of NAT ^omega st ( x = qN & qN in Domin_0 (n,m) & 2 * (j + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (qN | N)) = N & N > 0 ) } ) by A18; A20: x in C . i by A17, XBOOLE_0:def_4; C . i = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (i + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } by A3, A14; then ex pN being Element of NAT ^omega st ( x = pN & pN in Domin_0 (n,m) & 2 * (i + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) by A20; hence contradiction by A16, A19; ::_thesis: verum end; A21: for k being Nat st k in dom C holds C . k is finite proof let k be Nat; ::_thesis: ( k in dom C implies C . k is finite ) assume k in dom C ; ::_thesis: C . k is finite then A22: C . k in rng C by FUNCT_1:3; thus C . k is finite by A22; ::_thesis: verum end; consider CardC being XFinSequence of such that A23: dom CardC = dom C and A24: for i being Nat st i in dom CardC holds CardC . i = card (C . i) and A25: card (union (rng C)) = Sum CardC by A21, A13, STIRL2_1:66; take CardC ; ::_thesis: ( card { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } = Sum CardC & dom CardC = m & ( for j being Nat st j < m holds CardC . j = (card (Domin_0 ((2 * j),j))) * (card (Domin_0 ((n -' (2 * (j + 1))),(m -' (j + 1))))) ) ) union (rng C) c= { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union (rng C) or x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } ) assume x in union (rng C) ; ::_thesis: x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } then consider y being set such that A26: x in y and A27: y in rng C by TARSKI:def_4; consider j being set such that A28: j in dom C and A29: C . j = y by A27, FUNCT_1:def_3; reconsider j = j as Element of NAT by A28; y = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (j + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } by A3, A28, A29; then consider pN being Element of NAT ^omega such that A30: ( x = pN & pN in Domin_0 (n,m) & 2 * (j + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) by A26; 2 * (j + 1) <> 0 ; then { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} by A30, NAT_1:def_1; hence x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } by A30; ::_thesis: verum end; hence ( card { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } = Sum CardC & dom CardC = m ) by A3, A23, A25, A4, XBOOLE_0:def_10; ::_thesis: for j being Nat st j < m holds CardC . j = (card (Domin_0 ((2 * j),j))) * (card (Domin_0 ((n -' (2 * (j + 1))),(m -' (j + 1))))) let j be Nat; ::_thesis: ( j < m implies CardC . j = (card (Domin_0 ((2 * j),j))) * (card (Domin_0 ((n -' (2 * (j + 1))),(m -' (j + 1))))) ) assume A31: j < m ; ::_thesis: CardC . j = (card (Domin_0 ((2 * j),j))) * (card (Domin_0 ((n -' (2 * (j + 1))),(m -' (j + 1))))) A32: m >= j + 1 by A31, NAT_1:13; then A33: m -' (j + 1) = m - (j + 1) by XREAL_1:233; set P = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (j + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } ; A34: j in dom C by A3, A31, NAT_1:44; then A35: C . j = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & 2 * (j + 1) = min* { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ) } by A3; 2 * (j + 1) <= 2 * m by A32, XREAL_1:64; then A36: n -' (2 * (j + 1)) = n - (2 * (j + 1)) by A1, XREAL_1:233, XXREAL_0:2; CardC . j = card (C . j) by A23, A24, A34; hence CardC . j = (card (Domin_0 ((2 * j),j))) * (card (Domin_0 ((n -' (2 * (j + 1))),(m -' (j + 1))))) by A36, A33, A35, Th36; ::_thesis: verum end; theorem Th38: :: CATALAN2:38 for n being Nat st n > 0 holds Domin_0 ((2 * n),n) = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((2 * n),n) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } proof let n be Nat; ::_thesis: ( n > 0 implies Domin_0 ((2 * n),n) = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((2 * n),n) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } ) assume A1: n > 0 ; ::_thesis: Domin_0 ((2 * n),n) = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((2 * n),n) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } set P = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((2 * n),n) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } ; thus Domin_0 ((2 * n),n) c= { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((2 * n),n) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } :: according to XBOOLE_0:def_10 ::_thesis: { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((2 * n),n) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } c= Domin_0 ((2 * n),n) proof A2: n + n > 0 + 0 by A1; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Domin_0 ((2 * n),n) or x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((2 * n),n) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } ) assume A3: x in Domin_0 ((2 * n),n) ; ::_thesis: x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((2 * n),n) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } consider p being XFinSequence of such that A4: x = p and p is dominated_by_0 and A5: ( dom p = 2 * n & Sum p = n ) by A3, Def2; A6: p in NAT ^omega by AFINSQ_1:def_7; 2 * (Sum (p | (2 * n))) = 2 * n by A5, RELAT_1:69; then 2 * n in { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } by A2; hence x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((2 * n),n) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } by A3, A4, A6; ::_thesis: verum end; thus { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((2 * n),n) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } c= Domin_0 ((2 * n),n) ::_thesis: verum proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((2 * n),n) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } or x in Domin_0 ((2 * n),n) ) assume x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((2 * n),n) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } ; ::_thesis: x in Domin_0 ((2 * n),n) then ex pN being Element of NAT ^omega st ( x = pN & pN in Domin_0 ((2 * n),n) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) ; hence x in Domin_0 ((2 * n),n) ; ::_thesis: verum end; end; theorem Th39: :: CATALAN2:39 for n being Nat st n > 0 holds ex Catal being XFinSequence of st ( Sum Catal = Catalan (n + 1) & dom Catal = n & ( for j being Nat st j < n holds Catal . j = (Catalan (j + 1)) * (Catalan (n -' j)) ) ) proof let n be Nat; ::_thesis: ( n > 0 implies ex Catal being XFinSequence of st ( Sum Catal = Catalan (n + 1) & dom Catal = n & ( for j being Nat st j < n holds Catal . j = (Catalan (j + 1)) * (Catalan (n -' j)) ) ) ) assume A1: n > 0 ; ::_thesis: ex Catal being XFinSequence of st ( Sum Catal = Catalan (n + 1) & dom Catal = n & ( for j being Nat st j < n holds Catal . j = (Catalan (j + 1)) * (Catalan (n -' j)) ) ) consider CardF being XFinSequence of such that A2: card { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((2 * n),n) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } = Sum CardF and A3: dom CardF = n and A4: for j being Nat st j < n holds CardF . j = (card (Domin_0 ((2 * j),j))) * (card (Domin_0 (((2 * n) -' (2 * (j + 1))),(n -' (j + 1))))) by Th37; take CardF ; ::_thesis: ( Sum CardF = Catalan (n + 1) & dom CardF = n & ( for j being Nat st j < n holds CardF . j = (Catalan (j + 1)) * (Catalan (n -' j)) ) ) Sum CardF = card (Domin_0 ((2 * n),n)) by A1, A2, Th38; hence ( Sum CardF = Catalan (n + 1) & dom CardF = n ) by A3, Th34; ::_thesis: for j being Nat st j < n holds CardF . j = (Catalan (j + 1)) * (Catalan (n -' j)) let j be Nat; ::_thesis: ( j < n implies CardF . j = (Catalan (j + 1)) * (Catalan (n -' j)) ) assume A5: j < n ; ::_thesis: CardF . j = (Catalan (j + 1)) * (Catalan (n -' j)) n - j > j - j by A5, XREAL_1:9; then n -' j > 0 by A5, XREAL_1:233; then reconsider nj = (n -' j) - 1 as Element of NAT by NAT_1:20; j + 1 <= n by A5, NAT_1:13; then A6: ( (2 * n) -' (2 * (j + 1)) = (2 * n) - (2 * (j + 1)) & n -' (j + 1) = n - (j + 1) ) by XREAL_1:64, XREAL_1:233; A7: card (Domin_0 ((2 * j),j)) = Catalan (j + 1) by Th34; n - j = n -' j by A5, XREAL_1:233; then card (Domin_0 (((2 * n) -' (2 * (j + 1))),(n -' (j + 1)))) = card (Domin_0 ((2 * nj),nj)) by A6 .= Catalan (nj + 1) by Th34 ; hence CardF . j = (Catalan (j + 1)) * (Catalan (n -' j)) by A4, A5, A7; ::_thesis: verum end; theorem Th40: :: CATALAN2:40 for n, m being Nat holds card { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } = card (Domin_0 (n,m)) proof let n, m be Nat; ::_thesis: card { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } = card (Domin_0 (n,m)) defpred S1[ set , set ] means ex p being XFinSequence of st ( $1 = <%0%> ^ p & $2 = p ); set F = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } ; set Z = Domin_0 (n,m); A1: for x being set st x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } holds ex y being set st ( y in Domin_0 (n,m) & S1[x,y] ) proof A2: len <%0%> = 1 by AFINSQ_1:33; let x be set ; ::_thesis: ( x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } implies ex y being set st ( y in Domin_0 (n,m) & S1[x,y] ) ) A3: Sum <%0%> = 0 by AFINSQ_2:53; assume x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } ; ::_thesis: ex y being set st ( y in Domin_0 (n,m) & S1[x,y] ) then consider pN being Element of NAT ^omega such that A4: ( x = pN & pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) ; A5: len pN = dom pN ; ( pN is dominated_by_0 & dom pN = n + 1 ) by A4, Th20; then consider q being XFinSequence of such that A6: pN = <%0%> ^ q and A7: q is dominated_by_0 by A4, A5, Th17; dom pN = (len <%0%>) + (len q) by A6, AFINSQ_1:def_3; then A8: n + 1 = (len q) + 1 by A4, A2, Th20; take q ; ::_thesis: ( q in Domin_0 (n,m) & S1[x,q] ) Sum pN = (Sum <%0%>) + (Sum q) by A6, AFINSQ_2:55; then Sum q = m by A4, A3, Th20; hence ( q in Domin_0 (n,m) & S1[x,q] ) by A4, A6, A7, A8, Th20; ::_thesis: verum end; consider f being Function of { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } ,(Domin_0 (n,m)) such that A9: for x being set st x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } holds S1[x,f . x] from FUNCT_2:sch_1(A1); A10: ( Domin_0 (n,m) = {} implies { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } = {} ) proof assume Domin_0 (n,m) = {} ; ::_thesis: { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } = {} then 2 * m > n by Th22; then A11: 2 * m >= n + 1 by NAT_1:13; assume { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } <> {} ; ::_thesis: contradiction then consider x being set such that A12: x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } by XBOOLE_0:def_1; consider pN being Element of NAT ^omega such that A13: ( x = pN & pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) by A12; dom pN = n + 1 by A13, Th20; then pN | (n + 1) = pN by RELAT_1:69; then A14: Sum (pN | (n + 1)) = m by A13, Th20; pN is dominated_by_0 by A13, Th20; then 2 * m <= n + 1 by A14, Th2; then 2 * (Sum (pN | (n + 1))) = n + 1 by A14, A11, XXREAL_0:1; then n + 1 in { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } ; hence contradiction by A13; ::_thesis: verum end; then A15: dom f = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } by FUNCT_2:def_1; A16: rng f = Domin_0 (n,m) proof thus rng f c= Domin_0 (n,m) ; :: according to XBOOLE_0:def_10 ::_thesis: Domin_0 (n,m) c= rng f let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Domin_0 (n,m) or x in rng f ) assume x in Domin_0 (n,m) ; ::_thesis: x in rng f then consider p being XFinSequence of such that A17: p = x and A18: p is dominated_by_0 and A19: dom p = n and A20: Sum p = m by Def2; set q = <%0%> ^ p; A21: { N where N is Element of NAT : ( 2 * (Sum ((<%0%> ^ p) | N)) = N & N > 0 ) } = {} by A18, Th18; Sum (<%0%> ^ p) = (Sum <%0%>) + (Sum p) by AFINSQ_2:55; then A22: Sum (<%0%> ^ p) = 0 + m by A20, AFINSQ_2:53; A23: <%0%> ^ p in NAT ^omega by AFINSQ_1:def_7; dom (<%0%> ^ p) = (len <%0%>) + (len p) by AFINSQ_1:def_3; then A24: dom (<%0%> ^ p) = 1 + n by A19, AFINSQ_1:33; <%0%> ^ p is dominated_by_0 by A18, Th18; then <%0%> ^ p in Domin_0 ((n + 1),m) by A24, A22, Th20; then A25: <%0%> ^ p in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } by A21, A23; then consider r being XFinSequence of such that A26: <%0%> ^ p = <%0%> ^ r and A27: f . (<%0%> ^ p) = r by A9; r = p by A26, AFINSQ_1:28; hence x in rng f by A15, A17, A25, A27, FUNCT_1:3; ::_thesis: verum end; for x, y being set st x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } & y in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } & f . x = f . y holds x = y proof let x, y be set ; ::_thesis: ( x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } & y in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } & f . x = f . y implies x = y ) assume that A28: ( x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } & y in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } ) and A29: f . x = f . y ; ::_thesis: x = y ( ex p being XFinSequence of st ( x = <%0%> ^ p & f . x = p ) & ex q being XFinSequence of st ( y = <%0%> ^ q & f . y = q ) ) by A9, A28; hence x = y by A29; ::_thesis: verum end; then f is one-to-one by A10, FUNCT_2:19; then { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } , Domin_0 (n,m) are_equipotent by A15, A16, WELLORD2:def_4; hence card { pN where pN is Element of NAT ^omega : ( pN in Domin_0 ((n + 1),m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } = card (Domin_0 (n,m)) by CARD_1:5; ::_thesis: verum end; theorem :: CATALAN2:41 for n, m being Nat st 2 * m <= n holds ex CardF being XFinSequence of st ( card (Domin_0 (n,m)) = (Sum CardF) + (card (Domin_0 ((n -' 1),m))) & dom CardF = m & ( for j being Nat st j < m holds CardF . j = (card (Domin_0 ((2 * j),j))) * (card (Domin_0 ((n -' (2 * (j + 1))),(m -' (j + 1))))) ) ) proof let n, m be Nat; ::_thesis: ( 2 * m <= n implies ex CardF being XFinSequence of st ( card (Domin_0 (n,m)) = (Sum CardF) + (card (Domin_0 ((n -' 1),m))) & dom CardF = m & ( for j being Nat st j < m holds CardF . j = (card (Domin_0 ((2 * j),j))) * (card (Domin_0 ((n -' (2 * (j + 1))),(m -' (j + 1))))) ) ) ) assume A1: 2 * m <= n ; ::_thesis: ex CardF being XFinSequence of st ( card (Domin_0 (n,m)) = (Sum CardF) + (card (Domin_0 ((n -' 1),m))) & dom CardF = m & ( for j being Nat st j < m holds CardF . j = (card (Domin_0 ((2 * j),j))) * (card (Domin_0 ((n -' (2 * (j + 1))),(m -' (j + 1))))) ) ) set Z = Domin_0 (n,m); set Zne = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } ; A2: { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } c= Domin_0 (n,m) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } or x in Domin_0 (n,m) ) assume x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } ; ::_thesis: x in Domin_0 (n,m) then ex pN being Element of NAT ^omega st ( x = pN & pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) ; hence x in Domin_0 (n,m) ; ::_thesis: verum end; set Ze = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } ; A3: { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } c= Domin_0 (n,m) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } or x in Domin_0 (n,m) ) assume x in { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } ; ::_thesis: x in Domin_0 (n,m) then ex pN being Element of NAT ^omega st ( x = pN & pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) ; hence x in Domin_0 (n,m) ; ::_thesis: verum end; reconsider Zne = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) } as finite set by A2; consider C being XFinSequence of such that A4: card Zne = Sum C and A5: dom C = m and A6: for j being Nat st j < m holds C . j = (card (Domin_0 ((2 * j),j))) * (card (Domin_0 ((n -' (2 * (j + 1))),(m -' (j + 1))))) by A1, Th37; reconsider Ze = { pN where pN is Element of NAT ^omega : ( pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } = {} ) } as finite set by A3; take C ; ::_thesis: ( card (Domin_0 (n,m)) = (Sum C) + (card (Domin_0 ((n -' 1),m))) & dom C = m & ( for j being Nat st j < m holds C . j = (card (Domin_0 ((2 * j),j))) * (card (Domin_0 ((n -' (2 * (j + 1))),(m -' (j + 1))))) ) ) A7: Ze misses Zne proof assume Ze meets Zne ; ::_thesis: contradiction then consider x being set such that A8: x in Ze and A9: x in Zne by XBOOLE_0:3; A10: ex qN being Element of NAT ^omega st ( qN = x & qN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (qN | N)) = N & N > 0 ) } = {} ) by A8; ex pN being Element of NAT ^omega st ( pN = x & pN in Domin_0 (n,m) & { N where N is Element of NAT : ( 2 * (Sum (pN | N)) = N & N > 0 ) } <> {} ) by A9; hence contradiction by A10; ::_thesis: verum end; A11: Domin_0 (n,m) c= Ze \/ Zne proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Domin_0 (n,m) or x in Ze \/ Zne ) assume A12: x in Domin_0 (n,m) ; ::_thesis: x in Ze \/ Zne consider p being XFinSequence of such that A13: p = x and p is dominated_by_0 and dom p = n and Sum p = m by A12, Def2; reconsider p = p as Element of NAT ^omega by AFINSQ_1:def_7; set I = { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } ; now__::_thesis:_x_in_Ze_\/_Zne percases ( { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } = {} or { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } <> {} ) ; suppose { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } = {} ; ::_thesis: x in Ze \/ Zne then p in Ze by A12, A13; hence x in Ze \/ Zne by A13, XBOOLE_0:def_3; ::_thesis: verum end; suppose { N where N is Element of NAT : ( 2 * (Sum (p | N)) = N & N > 0 ) } <> {} ; ::_thesis: x in Ze \/ Zne then p in Zne by A12, A13; hence x in Ze \/ Zne by A13, XBOOLE_0:def_3; ::_thesis: verum end; end; end; hence x in Ze \/ Zne ; ::_thesis: verum end; Ze \/ Zne c= Domin_0 (n,m) by A3, A2, XBOOLE_1:8; then A14: Ze \/ Zne = Domin_0 (n,m) by A11, XBOOLE_0:def_10; now__::_thesis:_card_(Domin_0_(n,m))_=_(Sum_C)_+_(card_(Domin_0_((n_-'_1),m))) percases ( n = 0 or n > 0 ) ; supposeA15: n = 0 ; ::_thesis: card (Domin_0 (n,m)) = (Sum C) + (card (Domin_0 ((n -' 1),m))) then 2 * m = 0 by A1; then C = {} by A5; then A16: Sum C = 0 ; n - 1 < 1 - 1 by A15; hence card (Domin_0 (n,m)) = (Sum C) + (card (Domin_0 ((n -' 1),m))) by A15, A16, XREAL_0:def_2; ::_thesis: verum end; supposeA17: n > 0 ; ::_thesis: card (Domin_0 (n,m)) = (Sum C) + (card (Domin_0 ((n -' 1),m))) then reconsider n1 = n - 1 as Element of NAT by NAT_1:20; n = n1 + 1 ; then A18: card Ze = card (Domin_0 (n1,m)) by Th40; n1 = n -' 1 by A17, NAT_1:14, XREAL_1:233; hence card (Domin_0 (n,m)) = (Sum C) + (card (Domin_0 ((n -' 1),m))) by A7, A14, A4, A18, CARD_2:40; ::_thesis: verum end; end; end; hence ( card (Domin_0 (n,m)) = (Sum C) + (card (Domin_0 ((n -' 1),m))) & dom C = m & ( for j being Nat st j < m holds C . j = (card (Domin_0 ((2 * j),j))) * (card (Domin_0 ((n -' (2 * (j + 1))),(m -' (j + 1))))) ) ) by A5, A6; ::_thesis: verum end; theorem :: CATALAN2:42 for n, k being Nat ex p being XFinSequence of st ( Sum p = card (Domin_0 ((((2 * n) + 2) + k),(n + 1))) & dom p = k + 1 & ( for i being Nat st i <= k holds p . i = card (Domin_0 ((((2 * n) + 1) + i),n)) ) ) proof let n, k be Nat; ::_thesis: ex p being XFinSequence of st ( Sum p = card (Domin_0 ((((2 * n) + 2) + k),(n + 1))) & dom p = k + 1 & ( for i being Nat st i <= k holds p . i = card (Domin_0 ((((2 * n) + 1) + i),n)) ) ) defpred S1[ set , set ] means for j being Nat st $1 = j holds $2 = card (Domin_0 ((((2 * n) + 1) + j),n)); A1: for i being Nat st i in k + 1 holds ex x being Element of NAT st S1[i,x] proof let i be Nat; ::_thesis: ( i in k + 1 implies ex x being Element of NAT st S1[i,x] ) assume i in k + 1 ; ::_thesis: ex x being Element of NAT st S1[i,x] S1[i, card (Domin_0 ((((2 * n) + 1) + i),n))] ; hence ex x being Element of NAT st S1[i,x] ; ::_thesis: verum end; consider p being XFinSequence of such that A2: dom p = k + 1 and A3: for i being Nat st i in k + 1 holds S1[i,p . i] from STIRL2_1:sch_5(A1); take p ; ::_thesis: ( Sum p = card (Domin_0 ((((2 * n) + 2) + k),(n + 1))) & dom p = k + 1 & ( for i being Nat st i <= k holds p . i = card (Domin_0 ((((2 * n) + 1) + i),n)) ) ) A4: for i being Nat st i <= k holds p . i = card (Domin_0 ((((2 * n) + 1) + i),n)) proof let i be Nat; ::_thesis: ( i <= k implies p . i = card (Domin_0 ((((2 * n) + 1) + i),n)) ) assume i <= k ; ::_thesis: p . i = card (Domin_0 ((((2 * n) + 1) + i),n)) then i < k + 1 by NAT_1:13; then i in k + 1 by NAT_1:44; hence p . i = card (Domin_0 ((((2 * n) + 1) + i),n)) by A3; ::_thesis: verum end; now__::_thesis:_Sum_p_=_card_(Domin_0_((((2_*_n)_+_2)_+_k),(n_+_1))) percases ( n = 0 or n > 0 ) ; supposeA5: n = 0 ; ::_thesis: Sum p = card (Domin_0 ((((2 * n) + 2) + k),(n + 1))) for x being set st x in dom p holds p . x = 1 proof let x be set ; ::_thesis: ( x in dom p implies p . x = 1 ) assume A6: x in dom p ; ::_thesis: p . x = 1 reconsider i = x as Element of NAT by A6; p . i = card (Domin_0 ((((2 * n) + 1) + i),n)) by A2, A3, A6; hence p . x = 1 by A5, Th24; ::_thesis: verum end; then p = (k + 1) --> 1 by A2, FUNCOP_1:11; then Sum p = (k + 1) * 1 by AFINSQ_2:58; hence Sum p = card (Domin_0 ((((2 * n) + 2) + k),(n + 1))) by A5, Th30; ::_thesis: verum end; suppose n > 0 ; ::_thesis: Sum p = card (Domin_0 ((((2 * n) + 2) + k),(n + 1))) then reconsider n1 = n - 1 as Element of NAT by NAT_1:20; defpred S2[ Nat] means for q being XFinSequence of st dom q = $1 + 1 & ( for i being Nat st i <= $1 holds q . i = card (Domin_0 ((((2 * n) + 1) + i),n)) ) holds Sum q = card (Domin_0 ((((2 * n) + 2) + $1),(n + 1))); A7: for j being Nat st S2[j] holds S2[j + 1] proof let j be Nat; ::_thesis: ( S2[j] implies S2[j + 1] ) assume A8: S2[j] ; ::_thesis: S2[j + 1] set CH2 = (((2 * n) + 2) + j) choose (n1 + 1); set CH1 = (((2 * n) + 2) + j) choose (n + 1); set j1 = j + 1; let q be XFinSequence of ; ::_thesis: ( dom q = (j + 1) + 1 & ( for i being Nat st i <= j + 1 holds q . i = card (Domin_0 ((((2 * n) + 1) + i),n)) ) implies Sum q = card (Domin_0 ((((2 * n) + 2) + (j + 1)),(n + 1))) ) assume that A9: dom q = (j + 1) + 1 and A10: for i being Nat st i <= j + 1 holds q . i = card (Domin_0 ((((2 * n) + 1) + i),n)) ; ::_thesis: Sum q = card (Domin_0 ((((2 * n) + 2) + (j + 1)),(n + 1))) A11: 2 * (n + 1) <= (2 * (n + 1)) + (j + 1) by NAT_1:11; j + 1 <= (j + 1) + 1 by NAT_1:11; then j + 1 c= (j + 1) + 1 by NAT_1:39; then A12: dom (q | (j + 1)) = j + 1 by A9, RELAT_1:62; A13: for i being Nat st i <= j holds (q | (j + 1)) . i = card (Domin_0 ((((2 * n) + 1) + i),n)) proof let i be Nat; ::_thesis: ( i <= j implies (q | (j + 1)) . i = card (Domin_0 ((((2 * n) + 1) + i),n)) ) assume i <= j ; ::_thesis: (q | (j + 1)) . i = card (Domin_0 ((((2 * n) + 1) + i),n)) then i < j + 1 by NAT_1:13; then ( i in dom (q | (j + 1)) & q . i = card (Domin_0 ((((2 * n) + 1) + i),n)) ) by A10, A12, NAT_1:44; hence (q | (j + 1)) . i = card (Domin_0 ((((2 * n) + 1) + i),n)) by FUNCT_1:47; ::_thesis: verum end; set CH4 = (((2 * n) + 1) + (j + 1)) choose n1; set CH3 = (((2 * n) + 1) + (j + 1)) choose n; A14: ( 2 * n <= (2 * n) + (1 + (j + 1)) & n1 + 1 = n ) by NAT_1:11; q . (j + 1) = card (Domin_0 ((((2 * n) + 1) + (j + 1)),n)) by A10; then A15: q . (j + 1) = ((((2 * n) + 1) + (j + 1)) choose n) - ((((2 * n) + 1) + (j + 1)) choose n1) by A14, Th28; j + 1 < (j + 1) + 1 by NAT_1:13; then j + 1 in dom q by A9, NAT_1:44; then A16: Sum (q | ((j + 1) + 1)) = (Sum (q | (j + 1))) + (q . (j + 1)) by AFINSQ_2:65; 2 * (n + 1) <= ((2 * n) + 2) + j by NAT_1:11; then card (Domin_0 ((((2 * n) + 2) + j),(n + 1))) = ((((2 * n) + 2) + j) choose (n + 1)) - ((((2 * n) + 2) + j) choose (n1 + 1)) by Th28; then Sum (q | (j + 1)) = ((((2 * n) + 2) + j) choose (n + 1)) - ((((2 * n) + 2) + j) choose (n1 + 1)) by A8, A12, A13; then (Sum (q | (j + 1))) + (q . (j + 1)) = (((((2 * n) + 2) + j) choose (n + 1)) + ((((2 * n) + 2) + j) choose (n1 + 1))) - (((((2 * n) + 1) + (j + 1)) choose n) + ((((2 * n) + 1) + (j + 1)) choose n1)) by A15 .= (((((2 * n) + 2) + j) + 1) choose (n + 1)) - (((((2 * n) + 1) + (j + 1)) choose n) + ((((2 * n) + 1) + (j + 1)) choose n1)) by NEWTON:22 .= ((((2 * n) + 2) + (j + 1)) choose (n + 1)) - ((((2 * n) + 2) + (j + 1)) choose (n1 + 1)) by NEWTON:22 .= card (Domin_0 ((((2 * n) + 2) + (j + 1)),(n + 1))) by A11, Th28 ; hence Sum q = card (Domin_0 ((((2 * n) + 2) + (j + 1)),(n + 1))) by A9, A16, RELAT_1:69; ::_thesis: verum end; A17: S2[ 0 ] proof reconsider 2n1 = (2 * n) + 1 as Element of NAT ; set 2CHn = ((2 * n) + 2) choose n; set 2CHn91 = ((2 * n) + 2) choose (n + 1); set CHn91 = 2n1 choose (n + 1); set CHn1 = 2n1 choose n1; set CHn = 2n1 choose n; let q be XFinSequence of ; ::_thesis: ( dom q = 0 + 1 & ( for i being Nat st i <= 0 holds q . i = card (Domin_0 ((((2 * n) + 1) + i),n)) ) implies Sum q = card (Domin_0 ((((2 * n) + 2) + 0),(n + 1))) ) assume ( dom q = 0 + 1 & ( for i being Nat st i <= 0 holds q . i = card (Domin_0 ((((2 * n) + 1) + i),n)) ) ) ; ::_thesis: Sum q = card (Domin_0 ((((2 * n) + 2) + 0),(n + 1))) then A18: ( q . 0 = card (Domin_0 ((((2 * n) + 1) + 0),n)) & len q = 1 ) ; A19: (2 * n) + 2 = ((2 * n) + 1) + 1 ; then A20: ((2 * n) + 2) choose (n + 1) = (2n1 choose (n + 1)) + (2n1 choose n) by NEWTON:22; n1 + 1 = n ; then A21: ((2 * n) + 2) choose n = (2n1 choose n) + (2n1 choose n1) by A19, NEWTON:22; ( n <= n + (n + 1) & ((2 * n) + 1) - n = n + 1 ) by NAT_1:11; then A22: 2n1 choose n = 2n1 choose (n + 1) by NEWTON:20; 2 * (n + 1) = (2 * n) + 2 ; then A23: card (Domin_0 (((2 * n) + 2),(n + 1))) = (((2 * n) + 2) choose (n + 1)) - (((2 * n) + 2) choose n) by Th28; ( card (Domin_0 (2n1,(n1 + 1))) = (2n1 choose n) - (2n1 choose n1) & Sum <%(q . 0)%> = q . 0 ) by Th28, AFINSQ_2:53, NAT_1:11; hence Sum q = card (Domin_0 ((((2 * n) + 2) + 0),(n + 1))) by A20, A21, A22, A23, A18, AFINSQ_1:34; ::_thesis: verum end; for j being Nat holds S2[j] from NAT_1:sch_2(A17, A7); hence Sum p = card (Domin_0 ((((2 * n) + 2) + k),(n + 1))) by A2, A4; ::_thesis: verum end; end; end; hence ( Sum p = card (Domin_0 ((((2 * n) + 2) + k),(n + 1))) & dom p = k + 1 & ( for i being Nat st i <= k holds p . i = card (Domin_0 ((((2 * n) + 1) + i),n)) ) ) by A2, A4; ::_thesis: verum end; begin Lm3: for Fr being XFinSequence of st ( dom Fr = 1 or len Fr = 1 ) holds Sum Fr = Fr . 0 proof let Fr be XFinSequence of ; ::_thesis: ( ( dom Fr = 1 or len Fr = 1 ) implies Sum Fr = Fr . 0 ) assume ( dom Fr = 1 or len Fr = 1 ) ; ::_thesis: Sum Fr = Fr . 0 then len Fr = 1 ; then Fr = <%(Fr . 0)%> by AFINSQ_1:34; hence Sum Fr = Fr . 0 by AFINSQ_2:53; ::_thesis: verum end; Lm4: for Fr1, Fr2 being XFinSequence of st dom Fr1 = dom Fr2 & ( for n being Nat st n in len Fr1 holds Fr1 . n = Fr2 . ((len Fr1) -' (1 + n)) ) holds Sum Fr1 = Sum Fr2 proof let Fr1, Fr2 be XFinSequence of ; ::_thesis: ( dom Fr1 = dom Fr2 & ( for n being Nat st n in len Fr1 holds Fr1 . n = Fr2 . ((len Fr1) -' (1 + n)) ) implies Sum Fr1 = Sum Fr2 ) assume that A1: dom Fr1 = dom Fr2 and A2: for n being Nat st n in len Fr1 holds Fr1 . n = Fr2 . ((len Fr1) -' (1 + n)) ; ::_thesis: Sum Fr1 = Sum Fr2 defpred S1[ set , set ] means for i being Nat st i = $1 holds $2 = (len Fr1) -' (1 + i); A3: card (len Fr1) = card (len Fr1) ; A4: for x being set st x in len Fr1 holds ex y being set st ( y in len Fr1 & S1[x,y] ) proof let x be set ; ::_thesis: ( x in len Fr1 implies ex y being set st ( y in len Fr1 & S1[x,y] ) ) assume A5: x in len Fr1 ; ::_thesis: ex y being set st ( y in len Fr1 & S1[x,y] ) len Fr1 is Subset of NAT by STIRL2_1:8; then reconsider k = x as Element of NAT by A5; k < len Fr1 by A5, NAT_1:44; then k + 1 <= len Fr1 by NAT_1:13; then A6: (len Fr1) -' (1 + k) = (len Fr1) - (1 + k) by XREAL_1:233; take (len Fr1) -' (1 + k) ; ::_thesis: ( (len Fr1) -' (1 + k) in len Fr1 & S1[x,(len Fr1) -' (1 + k)] ) (len Fr1) + 0 < (len Fr1) + (1 + k) by XREAL_1:8; then (len Fr1) - (1 + k) < ((len Fr1) + (1 + k)) - (1 + k) by XREAL_1:9; hence ( (len Fr1) -' (1 + k) in len Fr1 & S1[x,(len Fr1) -' (1 + k)] ) by A6, NAT_1:44; ::_thesis: verum end; consider P being Function of (len Fr1),(len Fr1) such that A7: for x being set st x in len Fr1 holds S1[x,P . x] from FUNCT_2:sch_1(A4); A8: for x1, x2 being set st x1 in len Fr1 & x2 in len Fr1 & P . x1 = P . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in len Fr1 & x2 in len Fr1 & P . x1 = P . x2 implies x1 = x2 ) assume that A9: x1 in len Fr1 and A10: x2 in len Fr1 and A11: P . x1 = P . x2 ; ::_thesis: x1 = x2 len Fr1 is Subset of NAT by STIRL2_1:8; then reconsider i = x1, j = x2 as Element of NAT by A9, A10; j < len Fr1 by A10, NAT_1:44; then j + 1 <= len Fr1 by NAT_1:13; then (len Fr1) -' (1 + j) = (len Fr1) - (1 + j) by XREAL_1:233; then A12: P . x2 = (len Fr1) - (1 + j) by A7, A10; i < len Fr1 by A9, NAT_1:44; then i + 1 <= len Fr1 by NAT_1:13; then (len Fr1) -' (1 + i) = (len Fr1) - (1 + i) by XREAL_1:233; then P . x1 = (len Fr1) - (1 + i) by A7, A9; hence x1 = x2 by A11, A12; ::_thesis: verum end; then A13: P is one-to-one by FUNCT_2:56; P is one-to-one by A8, FUNCT_2:56; then P is onto by A3, STIRL2_1:60; then reconsider P = P as Permutation of (dom Fr1) by A13; A14: now__::_thesis:_for_x_being_set_st_x_in_dom_Fr1_holds_ Fr1_._x_=_Fr2_._(P_._x) let x be set ; ::_thesis: ( x in dom Fr1 implies Fr1 . x = Fr2 . (P . x) ) assume A15: x in dom Fr1 ; ::_thesis: Fr1 . x = Fr2 . (P . x) reconsider k = x as Element of NAT by A15; P . k = (len Fr1) -' (1 + k) by A7, A15; hence Fr1 . x = Fr2 . (P . x) by A2, A15; ::_thesis: verum end; A16: for x being set st x in dom Fr1 holds ( x in dom P & P . x in dom Fr2 ) by A1, FUNCT_2:52; for x being set st x in dom P & P . x in dom Fr2 holds x in dom Fr1 ; then Fr1 = Fr2 * P by A16, A14, FUNCT_1:10; then addreal "**" Fr1 = addreal "**" Fr2 by A1, AFINSQ_2:45 .= Sum Fr2 by AFINSQ_2:48 ; hence Sum Fr1 = Sum Fr2 by AFINSQ_2:48; ::_thesis: verum end; definition let seq1, seq2 be Real_Sequence; funcseq1 (##) seq2 -> Real_Sequence means :Def4: :: CATALAN2:def 4 for k being Nat ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) ) & Sum Fr = it . k ); existence ex b1 being Real_Sequence st for k being Nat ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) ) & Sum Fr = b1 . k ) proof defpred S1[ set , set ] means for k being Nat st k = $1 holds ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) ) & Sum Fr = $2 ); A1: for x being set st x in NAT holds ex y being set st ( y in REAL & S1[x,y] ) proof let x be set ; ::_thesis: ( x in NAT implies ex y being set st ( y in REAL & S1[x,y] ) ) assume x in NAT ; ::_thesis: ex y being set st ( y in REAL & S1[x,y] ) then reconsider k = x as Element of NAT ; defpred S2[ set , set ] means for i being Nat st i = $1 holds $2 = (seq1 . i) * (seq2 . (k -' i)); A2: for i being Nat st i in k + 1 holds ex z being Element of REAL st S2[i,z] proof let i be Nat; ::_thesis: ( i in k + 1 implies ex z being Element of REAL st S2[i,z] ) assume i in k + 1 ; ::_thesis: ex z being Element of REAL st S2[i,z] take (seq1 . i) * (seq2 . (k -' i)) ; ::_thesis: S2[i,(seq1 . i) * (seq2 . (k -' i))] thus S2[i,(seq1 . i) * (seq2 . (k -' i))] ; ::_thesis: verum end; consider Fr being XFinSequence of such that A3: dom Fr = k + 1 and A4: for i being Nat st i in k + 1 holds S2[i,Fr . i] from STIRL2_1:sch_5(A2); take Sum Fr ; ::_thesis: ( Sum Fr in REAL & S1[x, Sum Fr] ) for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) by A4; hence ( Sum Fr in REAL & S1[x, Sum Fr] ) by A3, XREAL_0:def_1; ::_thesis: verum end; consider seq3 being Real_Sequence such that A5: for x being set st x in NAT holds S1[x,seq3 . x] from FUNCT_2:sch_1(A1); for k being Nat ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) ) & Sum Fr = seq3 . k ) proof let k be Nat; ::_thesis: ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) ) & Sum Fr = seq3 . k ) k in NAT by ORDINAL1:def_12; hence ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) ) & Sum Fr = seq3 . k ) by A5; ::_thesis: verum end; hence ex b1 being Real_Sequence st for k being Nat ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) ) & Sum Fr = b1 . k ) ; ::_thesis: verum end; uniqueness for b1, b2 being Real_Sequence st ( for k being Nat ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) ) & Sum Fr = b1 . k ) ) & ( for k being Nat ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) ) & Sum Fr = b2 . k ) ) holds b1 = b2 proof let seq3, seq4 be Real_Sequence; ::_thesis: ( ( for k being Nat ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) ) & Sum Fr = seq3 . k ) ) & ( for k being Nat ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) ) & Sum Fr = seq4 . k ) ) implies seq3 = seq4 ) assume that A6: for k being Nat ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) ) & Sum Fr = seq3 . k ) and A7: for k being Nat ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) ) & Sum Fr = seq4 . k ) ; ::_thesis: seq3 = seq4 now__::_thesis:_for_x_being_set_st_x_in_NAT_holds_ seq3_._x_=_seq4_._x let x be set ; ::_thesis: ( x in NAT implies seq3 . x = seq4 . x ) assume x in NAT ; ::_thesis: seq3 . x = seq4 . x then reconsider k = x as Element of NAT ; consider Fr1 being XFinSequence of such that A8: dom Fr1 = k + 1 and A9: for n being Nat st n in k + 1 holds Fr1 . n = (seq1 . n) * (seq2 . (k -' n)) and A10: Sum Fr1 = seq3 . k by A6; consider Fr2 being XFinSequence of such that A11: dom Fr2 = k + 1 and A12: for n being Nat st n in k + 1 holds Fr2 . n = (seq1 . n) * (seq2 . (k -' n)) and A13: Sum Fr2 = seq4 . k by A7; now__::_thesis:_for_n_being_Nat_st_n_in_dom_Fr1_holds_ Fr1_._n_=_Fr2_._n let n be Nat; ::_thesis: ( n in dom Fr1 implies Fr1 . n = Fr2 . n ) assume A14: n in dom Fr1 ; ::_thesis: Fr1 . n = Fr2 . n Fr1 . n = (seq1 . n) * (seq2 . (k -' n)) by A8, A9, A14; hence Fr1 . n = Fr2 . n by A8, A12, A14; ::_thesis: verum end; hence seq3 . x = seq4 . x by A8, A10, A11, A13, AFINSQ_1:8; ::_thesis: verum end; hence seq3 = seq4 by FUNCT_2:12; ::_thesis: verum end; commutativity for b1, seq1, seq2 being Real_Sequence st ( for k being Nat ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) ) & Sum Fr = b1 . k ) ) holds for k being Nat ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq2 . n) * (seq1 . (k -' n)) ) & Sum Fr = b1 . k ) proof let seq3, seq1, seq2 be Real_Sequence; ::_thesis: ( ( for k being Nat ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) ) & Sum Fr = seq3 . k ) ) implies for k being Nat ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq2 . n) * (seq1 . (k -' n)) ) & Sum Fr = seq3 . k ) ) assume A15: for k being Nat ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) ) & Sum Fr = seq3 . k ) ; ::_thesis: for k being Nat ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq2 . n) * (seq1 . (k -' n)) ) & Sum Fr = seq3 . k ) let k be Nat; ::_thesis: ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq2 . n) * (seq1 . (k -' n)) ) & Sum Fr = seq3 . k ) consider Fr1 being XFinSequence of such that A16: dom Fr1 = k + 1 and A17: for n being Nat st n in k + 1 holds Fr1 . n = (seq1 . n) * (seq2 . (k -' n)) and A18: Sum Fr1 = seq3 . k by A15; defpred S1[ set , set ] means for i being Nat st i = $1 holds $2 = (seq2 . i) * (seq1 . (k -' i)); reconsider k9 = k as Element of NAT by ORDINAL1:def_12; A19: for i being Nat st i in k9 + 1 holds ex z being Element of REAL st S1[i,z] proof let i be Nat; ::_thesis: ( i in k9 + 1 implies ex z being Element of REAL st S1[i,z] ) assume i in k9 + 1 ; ::_thesis: ex z being Element of REAL st S1[i,z] take (seq2 . i) * (seq1 . (k -' i)) ; ::_thesis: S1[i,(seq2 . i) * (seq1 . (k -' i))] thus S1[i,(seq2 . i) * (seq1 . (k -' i))] ; ::_thesis: verum end; consider Fr2 being XFinSequence of such that A20: dom Fr2 = k9 + 1 and A21: for i being Nat st i in k9 + 1 holds S1[i,Fr2 . i] from STIRL2_1:sch_5(A19); take Fr2 ; ::_thesis: ( dom Fr2 = k + 1 & ( for n being Nat st n in k + 1 holds Fr2 . n = (seq2 . n) * (seq1 . (k -' n)) ) & Sum Fr2 = seq3 . k ) thus ( dom Fr2 = k + 1 & ( for n being Nat st n in k + 1 holds Fr2 . n = (seq2 . n) * (seq1 . (k -' n)) ) ) by A20, A21; ::_thesis: Sum Fr2 = seq3 . k now__::_thesis:_for_n_being_Nat_st_n_in_len_Fr1_holds_ Fr1_._n_=_Fr2_._((len_Fr1)_-'_(1_+_n)) let n be Nat; ::_thesis: ( n in len Fr1 implies Fr1 . n = Fr2 . ((len Fr1) -' (1 + n)) ) assume A22: n in len Fr1 ; ::_thesis: Fr1 . n = Fr2 . ((len Fr1) -' (1 + n)) A23: n < k + 1 by A16, A22, NAT_1:44; then n <= k by NAT_1:13; then A24: k -' n = k - n by XREAL_1:233; k -' n <= (k -' n) + n by NAT_1:11; then A25: k -' (k -' n) = k - (k -' n) by A24, XREAL_1:233; n + 1 <= len Fr2 by A20, A23, NAT_1:13; then A26: (len Fr2) -' (n + 1) = (k + 1) - (n + 1) by A20, XREAL_1:233; ( k - n <= k & k < k + 1 ) by NAT_1:13, XREAL_1:43; then k - n < k + 1 by XXREAL_0:2; then (len Fr2) -' (n + 1) in k + 1 by A26, NAT_1:44; then Fr2 . ((len Fr2) -' (n + 1)) = (seq2 . (k -' n)) * (seq1 . n) by A21, A26, A24, A25; hence Fr1 . n = Fr2 . ((len Fr1) -' (1 + n)) by A16, A17, A20, A22; ::_thesis: verum end; hence Sum Fr2 = seq3 . k by A16, A18, A20, Lm4; ::_thesis: verum end; end; :: deftheorem Def4 defines (##) CATALAN2:def_4_:_ for seq1, seq2, b3 being Real_Sequence holds ( b3 = seq1 (##) seq2 iff for k being Nat ex Fr being XFinSequence of st ( dom Fr = k + 1 & ( for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * (seq2 . (k -' n)) ) & Sum Fr = b3 . k ) ); theorem :: CATALAN2:43 for Fr1, Fr2 being XFinSequence of st dom Fr1 = dom Fr2 & ( for n being Nat st n in len Fr1 holds Fr1 . n = Fr2 . ((len Fr1) -' (1 + n)) ) holds Sum Fr1 = Sum Fr2 by Lm4; theorem Th44: :: CATALAN2:44 for r being real number for Fr1, Fr2 being XFinSequence of st dom Fr1 = dom Fr2 & ( for n being Nat st n in len Fr1 holds Fr1 . n = r * (Fr2 . n) ) holds Sum Fr1 = r * (Sum Fr2) proof let r be real number ; ::_thesis: for Fr1, Fr2 being XFinSequence of st dom Fr1 = dom Fr2 & ( for n being Nat st n in len Fr1 holds Fr1 . n = r * (Fr2 . n) ) holds Sum Fr1 = r * (Sum Fr2) let Fr1, Fr2 be XFinSequence of ; ::_thesis: ( dom Fr1 = dom Fr2 & ( for n being Nat st n in len Fr1 holds Fr1 . n = r * (Fr2 . n) ) implies Sum Fr1 = r * (Sum Fr2) ) assume that A1: dom Fr1 = dom Fr2 and A2: for n being Nat st n in len Fr1 holds Fr1 . n = r * (Fr2 . n) ; ::_thesis: Sum Fr1 = r * (Sum Fr2) A3: ( Fr1 | (dom Fr1) = Fr1 & Fr2 | (dom Fr1) = Fr2 ) by A1, RELAT_1:69; defpred S1[ Nat] means ( $1 <= len Fr1 implies Sum (Fr1 | $1) = r * (Sum (Fr2 | $1)) ); A4: for i being Nat st S1[i] holds S1[i + 1] proof let i be Nat; ::_thesis: ( S1[i] implies S1[i + 1] ) assume A5: S1[i] ; ::_thesis: S1[i + 1] assume A6: i + 1 <= len Fr1 ; ::_thesis: Sum (Fr1 | (i + 1)) = r * (Sum (Fr2 | (i + 1))) then i < len Fr1 by NAT_1:13; then A7: i in len Fr1 by NAT_1:44; then A8: Fr1 . i = r * (Fr2 . i) by A2; ( Sum (Fr1 | (i + 1)) = (Fr1 . i) + (Sum (Fr1 | i)) & Sum (Fr2 | (i + 1)) = (Fr2 . i) + (Sum (Fr2 | i)) ) by A1, A7, AFINSQ_2:65; hence Sum (Fr1 | (i + 1)) = r * (Sum (Fr2 | (i + 1))) by A5, A6, A8, NAT_1:13; ::_thesis: verum end; A9: S1[ 0 ] ; for i being Nat holds S1[i] from NAT_1:sch_2(A9, A4); hence Sum Fr1 = r * (Sum Fr2) by A3; ::_thesis: verum end; theorem :: CATALAN2:45 for seq1, seq2 being Real_Sequence for r being real number holds seq1 (##) (r (#) seq2) = r (#) (seq1 (##) seq2) proof let seq1, seq2 be Real_Sequence; ::_thesis: for r being real number holds seq1 (##) (r (#) seq2) = r (#) (seq1 (##) seq2) let r be real number ; ::_thesis: seq1 (##) (r (#) seq2) = r (#) (seq1 (##) seq2) set RS = r (#) seq2; set S = seq1 (##) seq2; now__::_thesis:_for_x_being_set_st_x_in_NAT_holds_ (seq1_(##)_(r_(#)_seq2))_._x_=_(r_(#)_(seq1_(##)_seq2))_._x let x be set ; ::_thesis: ( x in NAT implies (seq1 (##) (r (#) seq2)) . x = (r (#) (seq1 (##) seq2)) . x ) assume x in NAT ; ::_thesis: (seq1 (##) (r (#) seq2)) . x = (r (#) (seq1 (##) seq2)) . x then reconsider k = x as Element of NAT ; consider Fr1 being XFinSequence of such that A1: dom Fr1 = k + 1 and A2: for n being Nat st n in k + 1 holds Fr1 . n = (seq1 . n) * ((r (#) seq2) . (k -' n)) and A3: Sum Fr1 = (seq1 (##) (r (#) seq2)) . k by Def4; consider Fr2 being XFinSequence of such that A4: dom Fr2 = k + 1 and A5: for n being Nat st n in k + 1 holds Fr2 . n = (seq1 . n) * (seq2 . (k -' n)) and A6: Sum Fr2 = (seq1 (##) seq2) . k by Def4; now__::_thesis:_for_n_being_Nat_st_n_in_len_Fr1_holds_ Fr1_._n_=_r_*_(Fr2_._n) let n be Nat; ::_thesis: ( n in len Fr1 implies Fr1 . n = r * (Fr2 . n) ) assume n in len Fr1 ; ::_thesis: Fr1 . n = r * (Fr2 . n) then A7: ( Fr1 . n = (seq1 . n) * ((r (#) seq2) . (k -' n)) & Fr2 . n = (seq1 . n) * (seq2 . (k -' n)) ) by A1, A2, A5; (r (#) seq2) . (k -' n) = r * (seq2 . (k -' n)) by SEQ_1:9; hence Fr1 . n = r * (Fr2 . n) by A7; ::_thesis: verum end; then Sum Fr1 = r * (Sum Fr2) by A1, A4, Th44; hence (seq1 (##) (r (#) seq2)) . x = (r (#) (seq1 (##) seq2)) . x by A3, A6, SEQ_1:9; ::_thesis: verum end; hence seq1 (##) (r (#) seq2) = r (#) (seq1 (##) seq2) by FUNCT_2:12; ::_thesis: verum end; theorem :: CATALAN2:46 for seq1, seq2, seq3 being Real_Sequence holds seq1 (##) (seq2 + seq3) = (seq1 (##) seq2) + (seq1 (##) seq3) proof let seq1, seq2, seq3 be Real_Sequence; ::_thesis: seq1 (##) (seq2 + seq3) = (seq1 (##) seq2) + (seq1 (##) seq3) set S = seq2 + seq3; set S2 = seq1 (##) seq2; set S3 = seq1 (##) seq3; now__::_thesis:_for_x_being_set_st_x_in_NAT_holds_ (seq1_(##)_(seq2_+_seq3))_._x_=_((seq1_(##)_seq2)_+_(seq1_(##)_seq3))_._x let x be set ; ::_thesis: ( x in NAT implies (seq1 (##) (seq2 + seq3)) . x = ((seq1 (##) seq2) + (seq1 (##) seq3)) . x ) assume x in NAT ; ::_thesis: (seq1 (##) (seq2 + seq3)) . x = ((seq1 (##) seq2) + (seq1 (##) seq3)) . x then reconsider k = x as Element of NAT ; consider Fr being XFinSequence of such that A1: dom Fr = k + 1 and A2: for n being Nat st n in k + 1 holds Fr . n = (seq1 . n) * ((seq2 + seq3) . (k -' n)) and A3: Sum Fr = (seq1 (##) (seq2 + seq3)) . k by Def4; consider Fr1 being XFinSequence of such that A4: dom Fr1 = k + 1 and A5: for n being Nat st n in k + 1 holds Fr1 . n = (seq1 . n) * (seq2 . (k -' n)) and A6: Sum Fr1 = (seq1 (##) seq2) . k by Def4; A7: len Fr1 = len Fr by A1, A4; consider Fr2 being XFinSequence of such that A8: dom Fr2 = k + 1 and A9: for n being Nat st n in k + 1 holds Fr2 . n = (seq1 . n) * (seq3 . (k -' n)) and A10: Sum Fr2 = (seq1 (##) seq3) . k by Def4; A11: for n being Nat st n in dom Fr holds Fr . n = addreal . ((Fr1 . n),(Fr2 . n)) proof let n be Nat; ::_thesis: ( n in dom Fr implies Fr . n = addreal . ((Fr1 . n),(Fr2 . n)) ) assume A12: n in dom Fr ; ::_thesis: Fr . n = addreal . ((Fr1 . n),(Fr2 . n)) A13: Fr . n = (seq1 . n) * ((seq2 + seq3) . (k -' n)) by A1, A2, A12; A14: (seq2 + seq3) . (k -' n) = (seq2 . (k -' n)) + (seq3 . (k -' n)) by SEQ_1:7; ( Fr1 . n = (seq1 . n) * (seq2 . (k -' n)) & Fr2 . n = (seq1 . n) * (seq3 . (k -' n)) ) by A1, A5, A9, A12; then Fr . n = (Fr1 . n) + (Fr2 . n) by A13, A14; hence Fr . n = addreal . ((Fr1 . n),(Fr2 . n)) by BINOP_2:def_9; ::_thesis: verum end; len Fr1 = len Fr2 by A4, A8; then addreal "**" (Fr1 ^ Fr2) = addreal "**" Fr by A11, A7, AFINSQ_2:46; then Sum Fr = addreal "**" (Fr1 ^ Fr2) by AFINSQ_2:48; then Sum Fr = Sum (Fr1 ^ Fr2) by AFINSQ_2:48; then Sum Fr = (Sum Fr1) + (Sum Fr2) by AFINSQ_2:55; hence (seq1 (##) (seq2 + seq3)) . x = ((seq1 (##) seq2) + (seq1 (##) seq3)) . x by A3, A6, A10, SEQ_1:7; ::_thesis: verum end; hence seq1 (##) (seq2 + seq3) = (seq1 (##) seq2) + (seq1 (##) seq3) by FUNCT_2:12; ::_thesis: verum end; theorem Th47: :: CATALAN2:47 for seq1, seq2 being Real_Sequence holds (seq1 (##) seq2) . 0 = (seq1 . 0) * (seq2 . 0) proof let seq1, seq2 be Real_Sequence; ::_thesis: (seq1 (##) seq2) . 0 = (seq1 . 0) * (seq2 . 0) set S = (seq1 . 0) * (seq2 . 0); consider Fr being XFinSequence of such that A1: dom Fr = 0 + 1 and A2: for n being Nat st n in 0 + 1 holds Fr . n = (seq1 . n) * (seq2 . (0 -' n)) and A3: Sum Fr = (seq1 (##) seq2) . 0 by Def4; A4: ( 0 -' 0 = 0 & len Fr = 1 ) by A1, XREAL_1:232; 0 in 1 by NAT_1:44; then Fr . 0 = (seq1 . 0) * (seq2 . (0 -' 0)) by A2; then Fr = <%((seq1 . 0) * (seq2 . 0))%> by A4, AFINSQ_1:34; hence (seq1 (##) seq2) . 0 = (seq1 . 0) * (seq2 . 0) by A3, AFINSQ_2:53; ::_thesis: verum end; theorem Th48: :: CATALAN2:48 for seq1, seq2 being Real_Sequence for n being Nat ex Fr being XFinSequence of st ( (Partial_Sums (seq1 (##) seq2)) . n = Sum Fr & dom Fr = n + 1 & ( for i being Nat st i in n + 1 holds Fr . i = (seq1 . i) * ((Partial_Sums seq2) . (n -' i)) ) ) proof let seq1, seq2 be Real_Sequence; ::_thesis: for n being Nat ex Fr being XFinSequence of st ( (Partial_Sums (seq1 (##) seq2)) . n = Sum Fr & dom Fr = n + 1 & ( for i being Nat st i in n + 1 holds Fr . i = (seq1 . i) * ((Partial_Sums seq2) . (n -' i)) ) ) let n be Nat; ::_thesis: ex Fr being XFinSequence of st ( (Partial_Sums (seq1 (##) seq2)) . n = Sum Fr & dom Fr = n + 1 & ( for i being Nat st i in n + 1 holds Fr . i = (seq1 . i) * ((Partial_Sums seq2) . (n -' i)) ) ) set S = seq1 (##) seq2; set P = Partial_Sums seq2; defpred S1[ Nat] means ex Fr being XFinSequence of st ( (Partial_Sums (seq1 (##) seq2)) . $1 = Sum Fr & dom Fr = $1 + 1 & ( for i being Nat st i in $1 + 1 holds Fr . i = (seq1 . i) * ((Partial_Sums seq2) . ($1 -' i)) ) ); A1: for n being Nat st S1[n] holds S1[n + 1] proof set A = addreal ; let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] ) set n1 = n + 1; defpred S2[ set , set ] means for i being Nat st i = $1 holds $2 = (seq1 . i) * ((Partial_Sums seq2) . ((n + 1) -' i)); A2: ( (n + 1) -' (n + 1) = 0 & (Partial_Sums seq2) . 0 = seq2 . 0 ) by SERIES_1:def_1, XREAL_1:232; A3: for i being Nat st i in (n + 1) + 1 holds ex x being Element of REAL st S2[i,x] proof let i be Nat; ::_thesis: ( i in (n + 1) + 1 implies ex x being Element of REAL st S2[i,x] ) assume i in (n + 1) + 1 ; ::_thesis: ex x being Element of REAL st S2[i,x] take (seq1 . i) * ((Partial_Sums seq2) . ((n + 1) -' i)) ; ::_thesis: S2[i,(seq1 . i) * ((Partial_Sums seq2) . ((n + 1) -' i))] thus S2[i,(seq1 . i) * ((Partial_Sums seq2) . ((n + 1) -' i))] ; ::_thesis: verum end; consider Fr2 being XFinSequence of such that A4: dom Fr2 = (n + 1) + 1 and A5: for i being Nat st i in (n + 1) + 1 holds S2[i,Fr2 . i] from STIRL2_1:sch_5(A3); assume S1[n] ; ::_thesis: S1[n + 1] then consider Fr being XFinSequence of such that A6: (Partial_Sums (seq1 (##) seq2)) . n = Sum Fr and A7: dom Fr = n + 1 and A8: for i being Nat st i in n + 1 holds Fr . i = (seq1 . i) * ((Partial_Sums seq2) . (n -' i)) ; consider Fr1 being XFinSequence of such that A9: dom Fr1 = (n + 1) + 1 and A10: for i being Nat st i in (n + 1) + 1 holds Fr1 . i = (seq1 . i) * (seq2 . ((n + 1) -' i)) and A11: Sum Fr1 = (seq1 (##) seq2) . (n + 1) by Def4; A12: Fr1 | ((n + 1) + 1) = Fr1 by A9, RELAT_1:69; A13: for i being Nat st i in dom (Fr2 | (n + 1)) holds (Fr2 | (n + 1)) . i = addreal . ((Fr . i),((Fr1 | (n + 1)) . i)) proof let i be Nat; ::_thesis: ( i in dom (Fr2 | (n + 1)) implies (Fr2 | (n + 1)) . i = addreal . ((Fr . i),((Fr1 | (n + 1)) . i)) ) assume A14: i in dom (Fr2 | (n + 1)) ; ::_thesis: (Fr2 | (n + 1)) . i = addreal . ((Fr . i),((Fr1 | (n + 1)) . i)) A15: i in (dom Fr2) /\ (n + 1) by A14, RELAT_1:61; then i in dom (Fr1 | (n + 1)) by A9, A4, RELAT_1:61; then A16: Fr1 . i = (Fr1 | (n + 1)) . i by FUNCT_1:47; A17: i in n + 1 by A15, XBOOLE_0:def_4; then A18: i < n + 1 by NAT_1:44; then i <= n by NAT_1:13; then A19: n -' i = n - i by XREAL_1:233; ( i in (n + 1) + 1 & i in NAT ) by A4, A15, XBOOLE_0:def_4; then A20: ( Fr1 . i = (seq1 . i) * (seq2 . ((n + 1) -' i)) & Fr2 . i = (seq1 . i) * ((Partial_Sums seq2) . ((n + 1) -' i)) ) by A10, A5; A21: Fr2 . i = (Fr2 | (n + 1)) . i by A14, FUNCT_1:47; (n + 1) -' i = (n + 1) - i by A18, XREAL_1:233; then (n -' i) + 1 = (n + 1) -' i by A19; then A22: (Partial_Sums seq2) . ((n + 1) -' i) = ((Partial_Sums seq2) . (n -' i)) + (seq2 . ((n + 1) -' i)) by SERIES_1:def_1; Fr . i = (seq1 . i) * ((Partial_Sums seq2) . (n -' i)) by A8, A17; then Fr2 . i = (Fr . i) + (Fr1 . i) by A20, A22; hence (Fr2 | (n + 1)) . i = addreal . ((Fr . i),((Fr1 | (n + 1)) . i)) by A16, A21, BINOP_2:def_9; ::_thesis: verum end; n + 1 <= (n + 1) + 1 by NAT_1:11; then A23: n + 1 c= (n + 1) + 1 by NAT_1:39; then A24: len (Fr1 | (n + 1)) = len Fr by A7, A9, RELAT_1:62; n + 1 < (n + 1) + 1 by NAT_1:13; then A25: n + 1 in (n + 1) + 1 by NAT_1:44; then A26: ( Fr1 . (n + 1) = (seq1 . (n + 1)) * (seq2 . ((n + 1) -' (n + 1))) & Sum (Fr1 | ((n + 1) + 1)) = (Fr1 . (n + 1)) + (Sum (Fr1 | (n + 1))) ) by A9, A10, AFINSQ_2:65; len (Fr2 | (n + 1)) = len Fr by A7, A4, A23, RELAT_1:62; then addreal "**" (Fr2 | (n + 1)) = addreal "**" (Fr ^ (Fr1 | (n + 1))) by A13, A24, AFINSQ_2:46 .= Sum (Fr ^ (Fr1 | (n + 1))) by AFINSQ_2:48 .= (Sum Fr) + (Sum (Fr1 | (n + 1))) by AFINSQ_2:55 ; then A27: Sum (Fr2 | (n + 1)) = (Sum Fr) + (Sum (Fr1 | (n + 1))) by AFINSQ_2:48; take Fr2 ; ::_thesis: ( (Partial_Sums (seq1 (##) seq2)) . (n + 1) = Sum Fr2 & dom Fr2 = (n + 1) + 1 & ( for i being Nat st i in (n + 1) + 1 holds Fr2 . i = (seq1 . i) * ((Partial_Sums seq2) . ((n + 1) -' i)) ) ) ( Fr2 . (n + 1) = (seq1 . (n + 1)) * ((Partial_Sums seq2) . ((n + 1) -' (n + 1))) & Sum (Fr2 | ((n + 1) + 1)) = (Fr2 . (n + 1)) + (Sum (Fr2 | (n + 1))) ) by A4, A5, A25, AFINSQ_2:65; then ( Sum Fr2 = ((Partial_Sums (seq1 (##) seq2)) . n) + ((seq1 (##) seq2) . (n + 1)) & n in NAT & n + 1 in NAT ) by A6, A11, A4, A27, A2, A26, A12, ORDINAL1:def_12, RELAT_1:69; hence ( (Partial_Sums (seq1 (##) seq2)) . (n + 1) = Sum Fr2 & dom Fr2 = (n + 1) + 1 & ( for i being Nat st i in (n + 1) + 1 holds Fr2 . i = (seq1 . i) * ((Partial_Sums seq2) . ((n + 1) -' i)) ) ) by A4, A5, SERIES_1:def_1; ::_thesis: verum end; A28: S1[ 0 ] proof set Fr = 1 --> ((seq1 . 0) * (seq2 . 0)); reconsider Fr = 1 --> ((seq1 . 0) * (seq2 . 0)) as XFinSequence of ; take Fr ; ::_thesis: ( (Partial_Sums (seq1 (##) seq2)) . 0 = Sum Fr & dom Fr = 0 + 1 & ( for i being Nat st i in 0 + 1 holds Fr . i = (seq1 . i) * ((Partial_Sums seq2) . (0 -' i)) ) ) A29: dom Fr = 1 by FUNCOP_1:13; then A30: ( dom (Fr | 0) = 0 & Fr | 1 = Fr ) by RELAT_1:69; A31: 0 in 1 by NAT_1:44; then A32: Fr . 0 = (seq1 . 0) * (seq2 . 0) by FUNCOP_1:7; (Sum (Fr | 0)) + (Fr . 0) = Sum (Fr | (0 + 1)) by A29, A31, AFINSQ_2:65; then Sum Fr = (seq1 (##) seq2) . 0 by Th47, A32, A30; hence ( (Partial_Sums (seq1 (##) seq2)) . 0 = Sum Fr & dom Fr = 0 + 1 ) by FUNCOP_1:13, SERIES_1:def_1; ::_thesis: for i being Nat st i in 0 + 1 holds Fr . i = (seq1 . i) * ((Partial_Sums seq2) . (0 -' i)) let i be Nat; ::_thesis: ( i in 0 + 1 implies Fr . i = (seq1 . i) * ((Partial_Sums seq2) . (0 -' i)) ) assume A33: i in 0 + 1 ; ::_thesis: Fr . i = (seq1 . i) * ((Partial_Sums seq2) . (0 -' i)) i < 1 by A33, NAT_1:44; then A34: i = 0 by NAT_1:14; then 0 -' i = 0 by XREAL_1:232; then (Partial_Sums seq2) . (0 -' i) = seq2 . 0 by SERIES_1:def_1; hence Fr . i = (seq1 . i) * ((Partial_Sums seq2) . (0 -' i)) by A33, A34, FUNCOP_1:7; ::_thesis: verum end; for i being Nat holds S1[i] from NAT_1:sch_2(A28, A1); hence ex Fr being XFinSequence of st ( (Partial_Sums (seq1 (##) seq2)) . n = Sum Fr & dom Fr = n + 1 & ( for i being Nat st i in n + 1 holds Fr . i = (seq1 . i) * ((Partial_Sums seq2) . (n -' i)) ) ) ; ::_thesis: verum end; theorem Th49: :: CATALAN2:49 for seq1, seq2 being Real_Sequence for n being Nat st seq2 is summable holds ex Fr being XFinSequence of st ( (Partial_Sums (seq1 (##) seq2)) . n = ((Sum seq2) * ((Partial_Sums seq1) . n)) - (Sum Fr) & dom Fr = n + 1 & ( for i being Nat st i in n + 1 holds Fr . i = (seq1 . i) * (Sum (seq2 ^\ ((n -' i) + 1))) ) ) proof let seq1, seq2 be Real_Sequence; ::_thesis: for n being Nat st seq2 is summable holds ex Fr being XFinSequence of st ( (Partial_Sums (seq1 (##) seq2)) . n = ((Sum seq2) * ((Partial_Sums seq1) . n)) - (Sum Fr) & dom Fr = n + 1 & ( for i being Nat st i in n + 1 holds Fr . i = (seq1 . i) * (Sum (seq2 ^\ ((n -' i) + 1))) ) ) let n be Nat; ::_thesis: ( seq2 is summable implies ex Fr being XFinSequence of st ( (Partial_Sums (seq1 (##) seq2)) . n = ((Sum seq2) * ((Partial_Sums seq1) . n)) - (Sum Fr) & dom Fr = n + 1 & ( for i being Nat st i in n + 1 holds Fr . i = (seq1 . i) * (Sum (seq2 ^\ ((n -' i) + 1))) ) ) ) assume A1: seq2 is summable ; ::_thesis: ex Fr being XFinSequence of st ( (Partial_Sums (seq1 (##) seq2)) . n = ((Sum seq2) * ((Partial_Sums seq1) . n)) - (Sum Fr) & dom Fr = n + 1 & ( for i being Nat st i in n + 1 holds Fr . i = (seq1 . i) * (Sum (seq2 ^\ ((n -' i) + 1))) ) ) defpred S1[ set , set ] means for i being Nat st i = $1 holds $2 = (seq1 . i) * (Sum (seq2 ^\ ((n -' i) + 1))); set P2 = Partial_Sums seq2; set P1 = Partial_Sums seq1; set S = seq1 (##) seq2; A2: for i being Nat st i in n + 1 holds ex x being Element of REAL st S1[i,x] proof let i be Nat; ::_thesis: ( i in n + 1 implies ex x being Element of REAL st S1[i,x] ) assume i in n + 1 ; ::_thesis: ex x being Element of REAL st S1[i,x] take (seq1 . i) * (Sum (seq2 ^\ ((n -' i) + 1))) ; ::_thesis: S1[i,(seq1 . i) * (Sum (seq2 ^\ ((n -' i) + 1)))] thus S1[i,(seq1 . i) * (Sum (seq2 ^\ ((n -' i) + 1)))] ; ::_thesis: verum end; consider Fr being XFinSequence of such that A3: dom Fr = n + 1 and A4: for i being Nat st i in n + 1 holds S1[i,Fr . i] from STIRL2_1:sch_5(A2); consider Fr1 being XFinSequence of such that A5: (Partial_Sums (seq1 (##) seq2)) . n = Sum Fr1 and A6: dom Fr1 = n + 1 and A7: for i being Nat st i in n + 1 holds Fr1 . i = (seq1 . i) * ((Partial_Sums seq2) . (n -' i)) by Th48; A8: 0 in n + 1 by NAT_1:44; then A9: ( Fr1 . 0 = (seq1 . 0) * ((Partial_Sums seq2) . (n -' 0)) & Sum (Fr1 | (0 + 1)) = (Fr1 . 0) + (Sum (Fr1 | 0)) ) by A6, A7, AFINSQ_2:65; defpred S2[ Nat] means ( $1 + 1 <= n + 1 implies (Sum (Fr1 | ($1 + 1))) + (Sum (Fr | ($1 + 1))) = (Sum seq2) * ((Partial_Sums seq1) . $1) ); A10: for k being Nat st S2[k] holds S2[k + 1] proof let k be Nat; ::_thesis: ( S2[k] implies S2[k + 1] ) assume A11: S2[k] ; ::_thesis: S2[k + 1] reconsider k1 = k + 1 as Element of NAT ; assume A12: (k + 1) + 1 <= n + 1 ; ::_thesis: (Sum (Fr1 | ((k + 1) + 1))) + (Sum (Fr | ((k + 1) + 1))) = (Sum seq2) * ((Partial_Sums seq1) . (k + 1)) then k1 < n + 1 by NAT_1:13; then A13: k1 in n + 1 by NAT_1:44; then A14: ( Fr . k1 = (seq1 . k1) * (Sum (seq2 ^\ ((n -' k1) + 1))) & Sum (Fr1 | (k1 + 1)) = (Fr1 . k1) + (Sum (Fr1 | k1)) ) by A4, A6, AFINSQ_2:65; A15: (Sum (Fr1 | k1)) + (Sum (Fr | k1)) = (Sum seq2) * ((Partial_Sums seq1) . k) by A12, A11, NAT_1:13; A16: k in NAT by ORDINAL1:def_12; ( Sum (Fr | (k1 + 1)) = (Fr . k1) + (Sum (Fr | k1)) & Fr1 . k1 = (seq1 . k1) * ((Partial_Sums seq2) . (n -' k1)) ) by A3, A7, A13, AFINSQ_2:65; then (Sum (Fr | (k1 + 1))) + (Sum (Fr1 | (k1 + 1))) = ((seq1 . k1) * ((Sum (seq2 ^\ ((n -' k1) + 1))) + ((Partial_Sums seq2) . (n -' k1)))) + ((Sum seq2) * ((Partial_Sums seq1) . k)) by A15, A14 .= ((seq1 . k1) * (Sum seq2)) + ((Sum seq2) * ((Partial_Sums seq1) . k)) by A1, SERIES_1:15 .= (Sum seq2) * (((Partial_Sums seq1) . k) + (seq1 . k1)) .= ((Partial_Sums seq1) . k1) * (Sum seq2) by A16, SERIES_1:def_1 ; hence (Sum (Fr1 | ((k + 1) + 1))) + (Sum (Fr | ((k + 1) + 1))) = (Sum seq2) * ((Partial_Sums seq1) . (k + 1)) ; ::_thesis: verum end; ( Sum (Fr | (0 + 1)) = (Fr . 0) + (Sum (Fr | 0)) & Fr . 0 = (seq1 . 0) * (Sum (seq2 ^\ ((n -' 0) + 1))) ) by A3, A4, A8, AFINSQ_2:65; then (Sum (Fr | (0 + 1))) + (Sum (Fr1 | (0 + 1))) = (seq1 . 0) * ((Sum (seq2 ^\ ((n -' 0) + 1))) + ((Partial_Sums seq2) . (n -' 0))) by A9 .= (seq1 . 0) * (Sum seq2) by A1, SERIES_1:15 ; then A17: S2[ 0 ] by SERIES_1:def_1; A18: for k being Nat holds S2[k] from NAT_1:sch_2(A17, A10); take Fr ; ::_thesis: ( (Partial_Sums (seq1 (##) seq2)) . n = ((Sum seq2) * ((Partial_Sums seq1) . n)) - (Sum Fr) & dom Fr = n + 1 & ( for i being Nat st i in n + 1 holds Fr . i = (seq1 . i) * (Sum (seq2 ^\ ((n -' i) + 1))) ) ) A19: Fr1 | (n + 1) = Fr1 by A6, RELAT_1:69; Fr | (n + 1) = Fr by A3, RELAT_1:69; then (Sum Fr1) + (Sum Fr) = (Sum seq2) * ((Partial_Sums seq1) . n) by A18, A19; hence ( (Partial_Sums (seq1 (##) seq2)) . n = ((Sum seq2) * ((Partial_Sums seq1) . n)) - (Sum Fr) & dom Fr = n + 1 & ( for i being Nat st i in n + 1 holds Fr . i = (seq1 . i) * (Sum (seq2 ^\ ((n -' i) + 1))) ) ) by A3, A4, A5; ::_thesis: verum end; theorem Th50: :: CATALAN2:50 for Fr being XFinSequence of ex absFr being XFinSequence of st ( dom absFr = dom Fr & abs (Sum Fr) <= Sum absFr & ( for i being Nat st i in dom absFr holds absFr . i = abs (Fr . i) ) ) proof let Fr be XFinSequence of ; ::_thesis: ex absFr being XFinSequence of st ( dom absFr = dom Fr & abs (Sum Fr) <= Sum absFr & ( for i being Nat st i in dom absFr holds absFr . i = abs (Fr . i) ) ) defpred S1[ set , set ] means $2 = abs (Fr . $1); A1: Fr | (dom Fr) = Fr ; A2: for i being Nat st i in len Fr holds ex x being Element of REAL st S1[i,x] ; consider absFr being XFinSequence of such that A3: dom absFr = len Fr and A4: for i being Nat st i in len Fr holds S1[i,absFr . i] from STIRL2_1:sch_5(A2); defpred S2[ Nat] means ( $1 <= len Fr implies abs (Sum (Fr | $1)) <= Sum (absFr | $1) ); A5: for i being Nat st S2[i] holds S2[i + 1] proof let i be Nat; ::_thesis: ( S2[i] implies S2[i + 1] ) assume A6: S2[i] ; ::_thesis: S2[i + 1] set i1 = i + 1; assume A7: i + 1 <= len Fr ; ::_thesis: abs (Sum (Fr | (i + 1))) <= Sum (absFr | (i + 1)) then i < len Fr by NAT_1:13; then A8: i in dom Fr by NAT_1:44; then ( Sum (Fr | (i + 1)) = (Fr . i) + (Sum (Fr | i)) & absFr . i = abs (Fr . i) ) by A4, AFINSQ_2:65; then A9: abs (Sum (Fr | (i + 1))) <= (absFr . i) + (abs (Sum (Fr | i))) by COMPLEX1:56; Sum (absFr | (i + 1)) = (absFr . i) + (Sum (absFr | i)) by A3, A8, AFINSQ_2:65; then (absFr . i) + (abs (Sum (Fr | i))) <= Sum (absFr | (i + 1)) by A6, A7, NAT_1:13, XREAL_1:7; hence abs (Sum (Fr | (i + 1))) <= Sum (absFr | (i + 1)) by A9, XXREAL_0:2; ::_thesis: verum end; take absFr ; ::_thesis: ( dom absFr = dom Fr & abs (Sum Fr) <= Sum absFr & ( for i being Nat st i in dom absFr holds absFr . i = abs (Fr . i) ) ) A10: S2[ 0 ] by COMPLEX1:44; for i being Nat holds S2[i] from NAT_1:sch_2(A10, A5); then abs (Sum (Fr | (len Fr))) <= Sum (absFr | (len Fr)) ; hence ( dom absFr = dom Fr & abs (Sum Fr) <= Sum absFr & ( for i being Nat st i in dom absFr holds absFr . i = abs (Fr . i) ) ) by A3, A4, A1, RELAT_1:69; ::_thesis: verum end; theorem Th51: :: CATALAN2:51 for seq1 being Real_Sequence st seq1 is summable holds ex r being real number st ( 0 < r & ( for k being Nat holds abs (Sum (seq1 ^\ k)) < r ) ) proof let seq1 be Real_Sequence; ::_thesis: ( seq1 is summable implies ex r being real number st ( 0 < r & ( for k being Nat holds abs (Sum (seq1 ^\ k)) < r ) ) ) assume A1: seq1 is summable ; ::_thesis: ex r being real number st ( 0 < r & ( for k being Nat holds abs (Sum (seq1 ^\ k)) < r ) ) defpred S1[ Nat] means ex r being real number st ( r >= 0 & ( for i being Nat st i <= $1 holds abs (Sum (seq1 ^\ i)) <= r ) ); A2: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume S1[k] ; ::_thesis: S1[k + 1] then consider r being real number such that A3: r >= 0 and A4: for i being Nat st i <= k holds abs (Sum (seq1 ^\ i)) <= r ; take M = max (r,(abs (Sum (seq1 ^\ (k + 1))))); ::_thesis: ( M >= 0 & ( for i being Nat st i <= k + 1 holds abs (Sum (seq1 ^\ i)) <= M ) ) thus M >= 0 by A3, XXREAL_0:25; ::_thesis: for i being Nat st i <= k + 1 holds abs (Sum (seq1 ^\ i)) <= M let i be Nat; ::_thesis: ( i <= k + 1 implies abs (Sum (seq1 ^\ i)) <= M ) assume A5: i <= k + 1 ; ::_thesis: abs (Sum (seq1 ^\ i)) <= M now__::_thesis:_abs_(Sum_(seq1_^\_i))_<=_M percases ( i = k + 1 or i <= k ) by A5, NAT_1:8; suppose i = k + 1 ; ::_thesis: abs (Sum (seq1 ^\ i)) <= M hence abs (Sum (seq1 ^\ i)) <= M by XXREAL_0:25; ::_thesis: verum end; supposeA6: i <= k ; ::_thesis: abs (Sum (seq1 ^\ i)) <= M A7: r <= M by XXREAL_0:25; abs (Sum (seq1 ^\ i)) <= r by A4, A6; hence abs (Sum (seq1 ^\ i)) <= M by A7, XXREAL_0:2; ::_thesis: verum end; end; end; hence abs (Sum (seq1 ^\ i)) <= M ; ::_thesis: verum end; set P = Partial_Sums seq1; A8: lim (Partial_Sums seq1) = Sum seq1 by SERIES_1:def_3; Partial_Sums seq1 is convergent by A1, SERIES_1:def_2; then consider n being Element of NAT such that A9: for m being Element of NAT st n <= m holds abs (((Partial_Sums seq1) . m) - (Sum seq1)) < 1 by A8, SEQ_2:def_7; A10: S1[ 0 ] proof take abs (Sum seq1) ; ::_thesis: ( abs (Sum seq1) >= 0 & ( for i being Nat st i <= 0 holds abs (Sum (seq1 ^\ i)) <= abs (Sum seq1) ) ) thus abs (Sum seq1) >= 0 by COMPLEX1:46; ::_thesis: for i being Nat st i <= 0 holds abs (Sum (seq1 ^\ i)) <= abs (Sum seq1) let i be Nat; ::_thesis: ( i <= 0 implies abs (Sum (seq1 ^\ i)) <= abs (Sum seq1) ) assume i <= 0 ; ::_thesis: abs (Sum (seq1 ^\ i)) <= abs (Sum seq1) then i = 0 ; hence abs (Sum (seq1 ^\ i)) <= abs (Sum seq1) by NAT_1:47; ::_thesis: verum end; for k being Nat holds S1[k] from NAT_1:sch_2(A10, A2); then consider r being real number such that A11: r >= 0 and A12: for i being Nat st i <= n holds abs (Sum (seq1 ^\ i)) <= r ; take r1 = r + 1; ::_thesis: ( 0 < r1 & ( for k being Nat holds abs (Sum (seq1 ^\ k)) < r1 ) ) thus r1 > 0 by A11; ::_thesis: for k being Nat holds abs (Sum (seq1 ^\ k)) < r1 let k be Nat; ::_thesis: abs (Sum (seq1 ^\ k)) < r1 now__::_thesis:_abs_(Sum_(seq1_^\_k))_<_r1 percases ( k <= n or k > n ) ; supposeA13: k <= n ; ::_thesis: abs (Sum (seq1 ^\ k)) < r1 A14: 0 + r < r1 by XREAL_1:8; abs (Sum (seq1 ^\ k)) <= r by A12, A13; hence abs (Sum (seq1 ^\ k)) < r1 by A14, XXREAL_0:2; ::_thesis: verum end; supposeA15: k > n ; ::_thesis: abs (Sum (seq1 ^\ k)) < r1 then reconsider k1 = k - 1 as Element of NAT by NAT_1:20; k1 + 1 > n by A15; then k1 >= n by NAT_1:13; then A16: abs (((Partial_Sums seq1) . k1) - (Sum seq1)) < 1 by A9; Sum seq1 = ((Partial_Sums seq1) . k1) + (Sum (seq1 ^\ (k1 + 1))) by A1, SERIES_1:15; then abs (- (Sum (seq1 ^\ (k1 + 1)))) < 1 by A16; then A17: abs (Sum (seq1 ^\ (k1 + 1))) < 1 by COMPLEX1:52; 1 + 0 <= r1 by A11, XREAL_1:6; hence abs (Sum (seq1 ^\ k)) < r1 by A17, XXREAL_0:2; ::_thesis: verum end; end; end; hence abs (Sum (seq1 ^\ k)) < r1 ; ::_thesis: verum end; theorem Th52: :: CATALAN2:52 for seq1 being Real_Sequence for n, m being Nat st n <= m & ( for i being Nat holds seq1 . i >= 0 ) holds (Partial_Sums seq1) . n <= (Partial_Sums seq1) . m proof let seq1 be Real_Sequence; ::_thesis: for n, m being Nat st n <= m & ( for i being Nat holds seq1 . i >= 0 ) holds (Partial_Sums seq1) . n <= (Partial_Sums seq1) . m let n, m be Nat; ::_thesis: ( n <= m & ( for i being Nat holds seq1 . i >= 0 ) implies (Partial_Sums seq1) . n <= (Partial_Sums seq1) . m ) assume that A1: n <= m and A2: for i being Nat holds seq1 . i >= 0 ; ::_thesis: (Partial_Sums seq1) . n <= (Partial_Sums seq1) . m set S = Partial_Sums seq1; defpred S1[ Nat] means (Partial_Sums seq1) . n <= (Partial_Sums seq1) . (n + $1); A3: for i being Nat st S1[i] holds S1[i + 1] proof let i be Nat; ::_thesis: ( S1[i] implies S1[i + 1] ) assume A4: S1[i] ; ::_thesis: S1[i + 1] set ni = n + i; ( (Partial_Sums seq1) . ((n + i) + 1) = ((Partial_Sums seq1) . (n + i)) + (seq1 . ((n + i) + 1)) & seq1 . ((n + i) + 1) >= 0 ) by A2, SERIES_1:def_1; then (Partial_Sums seq1) . ((n + i) + 1) >= ((Partial_Sums seq1) . (n + i)) + 0 by XREAL_1:6; hence S1[i + 1] by A4, XXREAL_0:2; ::_thesis: verum end; A5: S1[ 0 ] ; A6: for i being Nat holds S1[i] from NAT_1:sch_2(A5, A3); reconsider m9 = m, n9 = n as Nat ; A7: n9 + (m9 - n9) = m9 ; m9 - n9 is Element of NAT by A1, NAT_1:21; hence (Partial_Sums seq1) . n <= (Partial_Sums seq1) . m by A6, A7; ::_thesis: verum end; theorem Th53: :: CATALAN2:53 for seq1, seq2 being Real_Sequence st seq1 is absolutely_summable & seq2 is summable holds ( seq1 (##) seq2 is summable & Sum (seq1 (##) seq2) = (Sum seq1) * (Sum seq2) ) proof let seq1, seq2 be Real_Sequence; ::_thesis: ( seq1 is absolutely_summable & seq2 is summable implies ( seq1 (##) seq2 is summable & Sum (seq1 (##) seq2) = (Sum seq1) * (Sum seq2) ) ) assume that A1: seq1 is absolutely_summable and A2: seq2 is summable ; ::_thesis: ( seq1 (##) seq2 is summable & Sum (seq1 (##) seq2) = (Sum seq1) * (Sum seq2) ) set S2 = Sum seq2; set S1 = Sum seq1; set PA = Partial_Sums (abs seq1); set P2 = Partial_Sums seq2; set P1 = Partial_Sums seq1; set S = seq1 (##) seq2; set P = Partial_Sums (seq1 (##) seq2); A3: for e being real number st 0 < e holds ex n being Element of NAT st for m being Element of NAT st n <= m holds abs (((Partial_Sums (seq1 (##) seq2)) . m) - ((Sum seq1) * (Sum seq2))) < e proof seq1 is summable by A1; then A4: Partial_Sums seq1 is convergent by SERIES_1:def_2; let e be real number ; ::_thesis: ( 0 < e implies ex n being Element of NAT st for m being Element of NAT st n <= m holds abs (((Partial_Sums (seq1 (##) seq2)) . m) - ((Sum seq1) * (Sum seq2))) < e ) assume A5: 0 < e ; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds abs (((Partial_Sums (seq1 (##) seq2)) . m) - ((Sum seq1) * (Sum seq2))) < e set e1 = e / (3 * ((abs (Sum seq2)) + 1)); (abs (Sum seq2)) + 1 > 0 + 0 by COMPLEX1:46, XREAL_1:8; then A6: 3 * ((abs (Sum seq2)) + 1) > 3 * 0 by XREAL_1:68; then ( lim (Partial_Sums seq1) = Sum seq1 & e / (3 * ((abs (Sum seq2)) + 1)) > 0 ) by A5, SERIES_1:def_3, XREAL_1:139; then consider n0 being Element of NAT such that A7: for n being Element of NAT st n0 <= n holds abs (((Partial_Sums seq1) . n) - (Sum seq1)) < e / (3 * ((abs (Sum seq2)) + 1)) by A4, SEQ_2:def_7; set e3 = e / (3 * ((Sum (abs seq1)) + 1)); A8: ( max (1,n0) = 1 or max (1,n0) = n0 ) by XXREAL_0:16; (abs (Sum seq2)) + 1 > 0 + 0 by COMPLEX1:46, XREAL_1:8; then A9: (e / (3 * ((abs (Sum seq2)) + 1))) * ((abs (Sum seq2)) + 1) = e / 3 by XCMPLX_1:92; A10: ( Partial_Sums seq2 is convergent & lim (Partial_Sums seq2) = Sum seq2 ) by A2, SERIES_1:def_2, SERIES_1:def_3; consider r being real number such that A11: 0 < r and A12: for k being Nat holds abs (Sum (seq2 ^\ k)) < r by A2, Th51; set e2 = e / (3 * r); A13: (abs (Sum seq2)) + 1 > (abs (Sum seq2)) + 0 by XREAL_1:8; A14: now__::_thesis:_for_n_being_Nat_holds_(abs_seq1)_._n_>=_0 let n be Nat; ::_thesis: (abs seq1) . n >= 0 n in NAT by ORDINAL1:def_12; then abs (seq1 . n) = (abs seq1) . n by SEQ_1:12; hence (abs seq1) . n >= 0 by COMPLEX1:46; ::_thesis: verum end; then A15: for n being Element of NAT holds (abs seq1) . n >= 0 ; A16: abs seq1 is summable by A1, SERIES_1:def_4; then Sum (abs seq1) >= 0 by A15, SERIES_1:18; then A17: ((Sum (abs seq1)) + 1) * (e / (3 * ((Sum (abs seq1)) + 1))) = e / 3 by XCMPLX_1:92; A18: Sum (abs seq1) >= 0 by A16, A15, SERIES_1:18; then 3 * ((Sum (abs seq1)) + 1) > 0 * 3 by XREAL_1:68; then e / (3 * ((Sum (abs seq1)) + 1)) > 0 by A5, XREAL_1:139; then consider n2 being Element of NAT such that A19: for n being Element of NAT st n2 <= n holds abs (((Partial_Sums seq2) . n) - (Sum seq2)) < e / (3 * ((Sum (abs seq1)) + 1)) by A10, SEQ_2:def_7; 3 * r > 0 * 3 by A11, XREAL_1:68; then e / (3 * r) > 0 by A5, XREAL_1:139; then consider n1 being Element of NAT such that A20: for n being Element of NAT st n1 <= n holds abs (((Partial_Sums (abs seq1)) . n) - ((Partial_Sums (abs seq1)) . n1)) < e / (3 * r) by A16, SERIES_1:21; ( max ((n1 + 1),n2) = n1 + 1 or max ((n1 + 1),n2) = n2 ) by XXREAL_0:16; then reconsider M = max ((max (1,n0)),(max ((n1 + 1),n2))) as Element of NAT by A8, XXREAL_0:16; A21: max ((n1 + 1),n2) <= M by XXREAL_0:25; take 2M = M * 2; ::_thesis: for m being Element of NAT st 2M <= m holds abs (((Partial_Sums (seq1 (##) seq2)) . m) - ((Sum seq1) * (Sum seq2))) < e let m be Element of NAT ; ::_thesis: ( 2M <= m implies abs (((Partial_Sums (seq1 (##) seq2)) . m) - ((Sum seq1) * (Sum seq2))) < e ) assume A22: 2M <= m ; ::_thesis: abs (((Partial_Sums (seq1 (##) seq2)) . m) - ((Sum seq1) * (Sum seq2))) < e A23: max (1,n0) <= M by XXREAL_0:25; then 0 < M by XXREAL_0:25; then reconsider M1 = M - 1 as Element of NAT by NAT_1:20; A24: M = M1 + 1 ; A25: n1 + 1 <= max ((n1 + 1),n2) by XXREAL_0:25; then M1 + 1 >= n1 + 1 by A21, XXREAL_0:2; then M1 >= n1 by XREAL_1:8; then (Partial_Sums (abs seq1)) . M1 >= (Partial_Sums (abs seq1)) . n1 by A14, Th52; then ((Partial_Sums (abs seq1)) . m) - ((Partial_Sums (abs seq1)) . M1) <= ((Partial_Sums (abs seq1)) . m) - ((Partial_Sums (abs seq1)) . n1) by XREAL_1:10; then A26: r * (((Partial_Sums (abs seq1)) . m) - ((Partial_Sums (abs seq1)) . M1)) <= r * (((Partial_Sums (abs seq1)) . m) - ((Partial_Sums (abs seq1)) . n1)) by A11, XREAL_1:64; consider Fr being XFinSequence of such that A27: (Partial_Sums (seq1 (##) seq2)) . m = ((Sum seq2) * ((Partial_Sums seq1) . m)) - (Sum Fr) and A28: dom Fr = m + 1 and A29: for i being Nat st i in m + 1 holds Fr . i = (seq1 . i) * (Sum (seq2 ^\ ((m -' i) + 1))) by A2, Th49; consider absFr being XFinSequence of such that A30: dom absFr = dom Fr and A31: abs (Sum Fr) <= Sum absFr and A32: for i being Nat st i in dom absFr holds absFr . i = abs (Fr . i) by Th50; A33: M <= M + M by NAT_1:11; then A34: M <= m by A22, XXREAL_0:2; then M < len absFr by A28, A30, NAT_1:13; then A35: len (absFr | M) = M by AFINSQ_1:11; n1 + 1 <= M by A25, A21, XXREAL_0:2; then n1 + 1 <= m by A34, XXREAL_0:2; then A36: n1 <= m by NAT_1:13; then (Partial_Sums (abs seq1)) . m >= (Partial_Sums (abs seq1)) . n1 by A14, Th52; then ((Partial_Sums (abs seq1)) . m) - ((Partial_Sums (abs seq1)) . n1) >= ((Partial_Sums (abs seq1)) . n1) - ((Partial_Sums (abs seq1)) . n1) by XREAL_1:9; then A37: abs (((Partial_Sums (abs seq1)) . m) - ((Partial_Sums (abs seq1)) . n1)) = ((Partial_Sums (abs seq1)) . m) - ((Partial_Sums (abs seq1)) . n1) by ABSVALUE:def_1; consider Fr1 being XFinSequence of such that A38: absFr = (absFr | M) ^ Fr1 by Th1; A39: m + 1 = (len (absFr | M)) + (len Fr1) by A28, A30, A38, AFINSQ_1:def_3; then A40: Fr1 | ((m - M) + 1) = Fr1 by A35, RELAT_1:69; A41: n2 <= max ((n1 + 1),n2) by XXREAL_0:25; then n2 <= M by A21, XXREAL_0:2; then n2 <= 2M by A33, XXREAL_0:2; then ( n2 <= m & m in NAT ) by A22, XXREAL_0:2; then A42: abs (((Partial_Sums seq2) . m) - (Sum seq2)) < e / (3 * ((Sum (abs seq1)) + 1)) by A19; defpred S1[ Nat] means ( (M + $1) + 1 <= m + 1 implies Sum (Fr1 | ($1 + 1)) <= r * (((Partial_Sums (abs seq1)) . (M + $1)) - ((Partial_Sums (abs seq1)) . M1)) ); A43: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A44: S1[k] ; ::_thesis: S1[k + 1] set k1 = k + 1; set Mk1 = M + (k + 1); A45: abs (seq1 . (M + (k + 1))) = (abs seq1) . (M + (k + 1)) by SEQ_1:12; assume A46: (M + (k + 1)) + 1 <= m + 1 ; ::_thesis: Sum (Fr1 | ((k + 1) + 1)) <= r * (((Partial_Sums (abs seq1)) . (M + (k + 1))) - ((Partial_Sums (abs seq1)) . M1)) then A47: M + (k + 1) < m + 1 by NAT_1:13; then A48: M + (k + 1) in m + 1 by NAT_1:44; then Fr . (M + (k + 1)) = (seq1 . (M + (k + 1))) * (Sum (seq2 ^\ ((m -' (M + (k + 1))) + 1))) by A29; then A49: abs (Fr . (M + (k + 1))) = (abs (seq1 . (M + (k + 1)))) * (abs (Sum (seq2 ^\ ((m -' (M + (k + 1))) + 1)))) by COMPLEX1:65; M + (k + 1) < m + 1 by A46, NAT_1:13; then k + 1 < len Fr1 by A39, A35, XREAL_1:7; then k + 1 in len Fr1 by NAT_1:44; then A50: Sum (Fr1 | ((k + 1) + 1)) = (Fr1 . (k + 1)) + (Sum (Fr1 | (k + 1))) by AFINSQ_2:65; m + 1 = len absFr by A28, A30; then absFr . (M + (k + 1)) = Fr1 . ((M + (k + 1)) - M) by A38, A35, A47, AFINSQ_1:19, NAT_1:11; then A51: Fr1 . (k + 1) = abs (Fr . (M + (k + 1))) by A28, A30, A32, A48; ( abs (seq1 . (M + (k + 1))) >= 0 & abs (Sum (seq2 ^\ ((m -' (M + (k + 1))) + 1))) < r ) by A12, COMPLEX1:46; then Fr1 . (k + 1) <= r * (abs (seq1 . (M + (k + 1)))) by A51, A49, XREAL_1:64; then Sum (Fr1 | ((k + 1) + 1)) <= (r * ((abs seq1) . (M + (k + 1)))) + (r * (((Partial_Sums (abs seq1)) . (M + k)) - ((Partial_Sums (abs seq1)) . M1))) by A44, A46, A50, A45, NAT_1:13, XREAL_1:7; then Sum (Fr1 | ((k + 1) + 1)) <= r * ((((abs seq1) . (M + (k + 1))) + ((Partial_Sums (abs seq1)) . (M + k))) - ((Partial_Sums (abs seq1)) . M1)) ; then Sum (Fr1 | ((k + 1) + 1)) <= r * (((Partial_Sums (abs seq1)) . ((M + k) + 1)) - ((Partial_Sums (abs seq1)) . M1)) by SERIES_1:def_1; hence Sum (Fr1 | ((k + 1) + 1)) <= r * (((Partial_Sums (abs seq1)) . (M + (k + 1))) - ((Partial_Sums (abs seq1)) . M1)) ; ::_thesis: verum end; abs (((Partial_Sums (abs seq1)) . m) - ((Partial_Sums (abs seq1)) . n1)) < e / (3 * r) by A20, A36; then r * (((Partial_Sums (abs seq1)) . m) - ((Partial_Sums (abs seq1)) . n1)) <= r * (e / (3 * r)) by A11, A37, XREAL_1:64; then A52: r * (((Partial_Sums (abs seq1)) . m) - ((Partial_Sums (abs seq1)) . M1)) <= r * (e / (3 * r)) by A26, XXREAL_0:2; A53: ( m = M + (m - M) & m - M = m -' M ) by A22, A33, XREAL_1:233, XXREAL_0:2; A54: S1[ 0 ] proof assume A55: (M + 0) + 1 <= m + 1 ; ::_thesis: Sum (Fr1 | (0 + 1)) <= r * (((Partial_Sums (abs seq1)) . (M + 0)) - ((Partial_Sums (abs seq1)) . M1)) then A56: M < m + 1 by NAT_1:13; then A57: M in m + 1 by NAT_1:44; then A58: Fr . M = (seq1 . M) * (Sum (seq2 ^\ ((m -' M) + 1))) by A29; (M + 1) - M <= (m + 1) - M by A55, XREAL_1:9; then 1 c= len Fr1 by A39, A35, NAT_1:39; then A59: dom (Fr1 | 1) = 1 by RELAT_1:62; m + 1 = len absFr by A28, A30; then absFr . M = Fr1 . (M - M) by A38, A35, A56, AFINSQ_1:19; then Fr1 . 0 = abs (Fr . M) by A28, A30, A32, A57; then A60: Fr1 . 0 = (abs (seq1 . M)) * (abs (Sum (seq2 ^\ ((m -' M) + 1)))) by A58, COMPLEX1:65; A61: ( abs (seq1 . M) >= 0 & r > abs (Sum (seq2 ^\ ((m -' M) + 1))) ) by A12, COMPLEX1:46; 0 in 1 by NAT_1:44; then A62: (Fr1 | 1) . 0 = Fr1 . 0 by A59, FUNCT_1:47; ((Partial_Sums (abs seq1)) . M1) + ((abs seq1) . (M1 + 1)) = (Partial_Sums (abs seq1)) . (M1 + 1) by SERIES_1:def_1; then A63: ((Partial_Sums (abs seq1)) . M) - ((Partial_Sums (abs seq1)) . M1) = abs (seq1 . M) by SEQ_1:12; Sum (Fr1 | 1) = (Fr1 | 1) . 0 by A59, Lm3; hence Sum (Fr1 | (0 + 1)) <= r * (((Partial_Sums (abs seq1)) . (M + 0)) - ((Partial_Sums (abs seq1)) . M1)) by A62, A60, A63, A61, XREAL_1:64; ::_thesis: verum end; for k being Nat holds S1[k] from NAT_1:sch_2(A54, A43); then Sum Fr1 <= r * (((Partial_Sums (abs seq1)) . m) - ((Partial_Sums (abs seq1)) . M1)) by A40, A53; then Sum Fr1 <= r * (e / (3 * r)) by A52, XXREAL_0:2; then A64: Sum Fr1 <= e / 3 by A11, XCMPLX_1:92; (abs seq1) . 0 >= 0 by A14; then A65: ((abs seq1) . 0) * (abs (((Partial_Sums seq2) . m) - (Sum seq2))) <= (e / (3 * ((Sum (abs seq1)) + 1))) * ((abs seq1) . 0) by A42, XREAL_1:64; A66: 0 in m + 1 by NAT_1:44; then A67: Fr . 0 = (seq1 . 0) * (Sum (seq2 ^\ ((m -' 0) + 1))) by A29; (Partial_Sums (abs seq1)) . M1 <= Sum (abs seq1) by A15, A16, RSSPACE2:3; then A68: (e / (3 * ((Sum (abs seq1)) + 1))) * ((Partial_Sums (abs seq1)) . M1) <= (e / (3 * ((Sum (abs seq1)) + 1))) * (Sum (abs seq1)) by A5, A18, XREAL_1:64; ( Sum seq2 = ((Partial_Sums seq2) . (m -' 0)) + (Sum (seq2 ^\ ((m -' 0) + 1))) & m -' 0 = m ) by A2, NAT_D:40, SERIES_1:15; then A69: Sum (seq2 ^\ ((m -' 0) + 1)) = (Sum seq2) - ((Partial_Sums seq2) . m) ; n0 <= max (1,n0) by XXREAL_0:25; then n0 <= M by A23, XXREAL_0:2; then ( n0 <= m & m in NAT ) by A34, XXREAL_0:2; then A70: abs (((Partial_Sums seq1) . m) - (Sum seq1)) < e / (3 * ((abs (Sum seq2)) + 1)) by A7; ( abs ((Sum seq2) * (((Partial_Sums seq1) . m) - (Sum seq1))) = (abs (Sum seq2)) * (abs (((Partial_Sums seq1) . m) - (Sum seq1))) & abs (Sum seq2) >= 0 ) by COMPLEX1:46, COMPLEX1:65; then A71: abs ((Sum seq2) * (((Partial_Sums seq1) . m) - (Sum seq1))) <= (abs (Sum seq2)) * (e / (3 * ((abs (Sum seq2)) + 1))) by A70, XREAL_1:64; A72: Sum absFr = (Sum (absFr | M)) + (Sum Fr1) by A38, AFINSQ_2:55; defpred S2[ Nat] means ( $1 + 1 <= M implies Sum (absFr | ($1 + 1)) <= (e / (3 * ((Sum (abs seq1)) + 1))) * ((Partial_Sums (abs seq1)) . $1) ); A73: n2 <= M by A41, A21, XXREAL_0:2; A74: for k being Nat st S2[k] holds S2[k + 1] proof let k be Nat; ::_thesis: ( S2[k] implies S2[k + 1] ) assume A75: S2[k] ; ::_thesis: S2[k + 1] reconsider k1 = k + 1 as Element of NAT ; A76: abs (seq1 . k1) = (abs seq1) . k1 by SEQ_1:12; A77: m - M >= 2M - M by A22, XREAL_1:9; assume A78: (k + 1) + 1 <= M ; ::_thesis: Sum (absFr | ((k + 1) + 1)) <= (e / (3 * ((Sum (abs seq1)) + 1))) * ((Partial_Sums (abs seq1)) . (k + 1)) then A79: k1 < M by NAT_1:13; then m - k1 >= m - M by XREAL_1:10; then m - k1 >= M by A77, XXREAL_0:2; then A80: m - k1 >= n2 by A73, XXREAL_0:2; ((e / (3 * ((Sum (abs seq1)) + 1))) * (abs (seq1 . k1))) + (Sum (absFr | k1)) <= ((e / (3 * ((Sum (abs seq1)) + 1))) * (abs (seq1 . k1))) + ((e / (3 * ((Sum (abs seq1)) + 1))) * ((Partial_Sums (abs seq1)) . k)) by A75, A78, NAT_1:13, XREAL_1:6; then ( ((e / (3 * ((Sum (abs seq1)) + 1))) * (abs (seq1 . k1))) + (Sum (absFr | k1)) <= (e / (3 * ((Sum (abs seq1)) + 1))) * (((abs seq1) . k1) + ((Partial_Sums (abs seq1)) . k)) & k in NAT ) by A76, ORDINAL1:def_12; then A81: ((e / (3 * ((Sum (abs seq1)) + 1))) * (abs (seq1 . k1))) + (Sum (absFr | k1)) <= (e / (3 * ((Sum (abs seq1)) + 1))) * ((Partial_Sums (abs seq1)) . k1) by SERIES_1:def_1; k1 < m by A34, A79, XXREAL_0:2; then k1 < m + 1 by NAT_1:13; then A82: k1 in m + 1 by NAT_1:44; then A83: Sum (absFr | (k1 + 1)) = (absFr . k1) + (Sum (absFr | k1)) by A28, A30, AFINSQ_2:65; m - k1 = m -' k1 by A34, A79, XREAL_1:233, XXREAL_0:2; then abs (((Partial_Sums seq2) . (m -' k1)) - (Sum seq2)) < e / (3 * ((Sum (abs seq1)) + 1)) by A19, A80; then A84: abs ((Sum seq2) - ((Partial_Sums seq2) . (m -' k1))) < e / (3 * ((Sum (abs seq1)) + 1)) by COMPLEX1:60; A85: Sum seq2 = ((Partial_Sums seq2) . (m -' k1)) + (Sum (seq2 ^\ ((m -' k1) + 1))) by A2, SERIES_1:15; abs (seq1 . k1) >= 0 by COMPLEX1:46; then (abs ((Sum seq2) - ((Partial_Sums seq2) . (m -' k1)))) * (abs (seq1 . k1)) <= (e / (3 * ((Sum (abs seq1)) + 1))) * (abs (seq1 . k1)) by A84, XREAL_1:64; then A86: abs ((seq1 . k1) * (Sum (seq2 ^\ ((m -' k1) + 1)))) <= (e / (3 * ((Sum (abs seq1)) + 1))) * (abs (seq1 . k1)) by A85, COMPLEX1:65; ( Fr . k1 = (seq1 . k1) * (Sum (seq2 ^\ ((m -' k1) + 1))) & abs (Fr . k1) = absFr . k1 ) by A28, A29, A30, A32, A82; then Sum (absFr | (k1 + 1)) <= ((e / (3 * ((Sum (abs seq1)) + 1))) * (abs (seq1 . k1))) + (Sum (absFr | k1)) by A83, A86, XREAL_1:6; hence Sum (absFr | ((k + 1) + 1)) <= (e / (3 * ((Sum (abs seq1)) + 1))) * ((Partial_Sums (abs seq1)) . (k + 1)) by A81, XXREAL_0:2; ::_thesis: verum end; ( Sum (absFr | (0 + 1)) = (absFr . 0) + (Sum (absFr | 0)) & absFr . 0 = abs (Fr . 0) ) by A28, A30, A32, A66, AFINSQ_2:65; then Sum (absFr | (0 + 1)) = (abs (seq1 . 0)) * (abs (Sum (seq2 ^\ ((m -' 0) + 1)))) by A67, COMPLEX1:65 .= ((abs seq1) . 0) * (abs (Sum (seq2 ^\ ((m -' 0) + 1)))) by SEQ_1:12 .= ((abs seq1) . 0) * (abs (((Partial_Sums seq2) . m) - (Sum seq2))) by A69, COMPLEX1:60 ; then A87: S2[ 0 ] by A65, SERIES_1:def_1; for k being Nat holds S2[k] from NAT_1:sch_2(A87, A74); then A88: Sum (absFr | M) <= (e / (3 * ((Sum (abs seq1)) + 1))) * ((Partial_Sums (abs seq1)) . M1) by A24; (Sum (abs seq1)) + 1 > (Sum (abs seq1)) + 0 by XREAL_1:8; then (e / (3 * ((Sum (abs seq1)) + 1))) * ((Sum (abs seq1)) + 1) >= (e / (3 * ((Sum (abs seq1)) + 1))) * (Sum (abs seq1)) by A5, A18, XREAL_1:64; then (e / (3 * ((Sum (abs seq1)) + 1))) * ((Partial_Sums (abs seq1)) . M1) <= e / 3 by A68, A17, XXREAL_0:2; then Sum (absFr | M) <= e / 3 by A88, XXREAL_0:2; then (Sum (absFr | M)) + (Sum Fr1) <= (e / 3) + (e / 3) by A64, XREAL_1:7; then A89: abs (Sum Fr) <= (e / 3) + (e / 3) by A31, A72, XXREAL_0:2; ((Partial_Sums (seq1 (##) seq2)) . m) - ((Sum seq1) * (Sum seq2)) = ((Sum seq2) * (((Partial_Sums seq1) . m) - (Sum seq1))) - (Sum Fr) by A27; then A90: abs (((Partial_Sums (seq1 (##) seq2)) . m) - ((Sum seq1) * (Sum seq2))) <= (abs ((Sum seq2) * (((Partial_Sums seq1) . m) - (Sum seq1)))) + (abs (Sum Fr)) by COMPLEX1:57; e / (3 * ((abs (Sum seq2)) + 1)) > 0 by A5, A6, XREAL_1:139; then (e / (3 * ((abs (Sum seq2)) + 1))) * ((abs (Sum seq2)) + 1) > (e / (3 * ((abs (Sum seq2)) + 1))) * (abs (Sum seq2)) by A13, XREAL_1:68; then abs ((Sum seq2) * (((Partial_Sums seq1) . m) - (Sum seq1))) < e / 3 by A9, A71, XXREAL_0:2; then (abs ((Sum seq2) * (((Partial_Sums seq1) . m) - (Sum seq1)))) + (abs (Sum Fr)) < (e / 3) + ((e / 3) + (e / 3)) by A89, XREAL_1:8; hence abs (((Partial_Sums (seq1 (##) seq2)) . m) - ((Sum seq1) * (Sum seq2))) < e by A90, XXREAL_0:2; ::_thesis: verum end; then A91: Partial_Sums (seq1 (##) seq2) is convergent by SEQ_2:def_6; hence seq1 (##) seq2 is summable by SERIES_1:def_2; ::_thesis: Sum (seq1 (##) seq2) = (Sum seq1) * (Sum seq2) lim (Partial_Sums (seq1 (##) seq2)) = (Sum seq1) * (Sum seq2) by A3, A91, SEQ_2:def_7; hence Sum (seq1 (##) seq2) = (Sum seq1) * (Sum seq2) by SERIES_1:def_3; ::_thesis: verum end; begin theorem :: CATALAN2:54 for r being real number ex Catal being Real_Sequence st ( ( for n being Nat holds Catal . n = (Catalan (n + 1)) * (r |^ n) ) & ( abs r < 1 / 4 implies ( Catal is absolutely_summable & Sum Catal = 1 + (r * ((Sum Catal) |^ 2)) & Sum Catal = 2 / (1 + (sqrt (1 - (4 * r)))) & ( r <> 0 implies Sum Catal = (1 - (sqrt (1 - (4 * r)))) / (2 * r) ) ) ) ) proof defpred S1[ set , set ] means for r being real number st $1 = r holds ex Catal being Real_Sequence st ( ( for n being Nat holds Catal . n = (Catalan (n + 1)) * (r |^ n) ) & ( abs r < 1 / 4 implies ( Catal is absolutely_summable & Sum Catal = 1 + (r * ((Sum Catal) |^ 2)) & $2 = Sum Catal ) ) ); A1: for x being set st x in REAL holds ex y being set st ( y in REAL & S1[x,y] ) proof let x be set ; ::_thesis: ( x in REAL implies ex y being set st ( y in REAL & S1[x,y] ) ) A2: abs 1 = 1 by ABSVALUE:def_1; assume x in REAL ; ::_thesis: ex y being set st ( y in REAL & S1[x,y] ) then reconsider r = x as Real ; set a = 4 * (abs r); deffunc H1( Element of NAT ) -> Element of REAL = (Catalan ($1 + 1)) * (r |^ $1); consider Cat being Real_Sequence such that A3: for n being Element of NAT holds Cat . n = H1(n) from FUNCT_2:sch_4(); set G = (4 * (abs r)) GeoSeq ; defpred S2[ Nat] means (abs Cat) . $1 <= ((4 * (abs r)) GeoSeq) . $1; A4: for n being Nat st S2[n] holds S2[n + 1] proof A5: abs r >= 0 by COMPLEX1:46; let n be Nat; ::_thesis: ( S2[n] implies S2[n + 1] ) assume S2[n] ; ::_thesis: S2[n + 1] then A6: (4 * (abs r)) * ((abs Cat) . n) <= (4 * (abs r)) * (((4 * (abs r)) GeoSeq) . n) by A5, XREAL_1:64; set n1 = n + 1; A7: ( n in NAT & n + 1 in NAT ) by ORDINAL1:def_12; A8: ( abs (r |^ (n + 1)) >= 0 & r |^ (n + 1) = r * (r |^ n) ) by COMPLEX1:46, NEWTON:6; Catalan ((n + 1) + 1) >= 0 by CATALAN1:17; then A9: abs (Catalan ((n + 1) + 1)) = Catalan ((n + 1) + 1) by ABSVALUE:def_1; Catalan (n + 1) >= 0 by CATALAN1:17; then abs (Catalan (n + 1)) = Catalan (n + 1) by ABSVALUE:def_1; then abs (Catalan ((n + 1) + 1)) < 4 * (abs (Catalan (n + 1))) by A9, CATALAN1:21; then A10: (abs (r |^ (n + 1))) * (abs (Catalan ((n + 1) + 1))) <= (4 * (abs (Catalan (n + 1)))) * (abs (r * (r |^ n))) by A8, XREAL_1:64; abs (r * (r |^ n)) = (abs r) * (abs (r |^ n)) by COMPLEX1:65; then abs ((r |^ (n + 1)) * (Catalan ((n + 1) + 1))) <= (4 * (abs r)) * ((abs (Catalan (n + 1))) * (abs (r |^ n))) by A10, COMPLEX1:65; then abs (Cat . (n + 1)) <= (4 * (abs r)) * ((abs (Catalan (n + 1))) * (abs (r |^ n))) by A3; then ( abs (Cat . (n + 1)) <= (4 * (abs r)) * (abs ((Catalan (n + 1)) * (r |^ n))) & n in NAT ) by COMPLEX1:65, ORDINAL1:def_12; then A11: abs (Cat . (n + 1)) <= (4 * (abs r)) * (abs (Cat . n)) by A3; abs (Cat . n) = (abs Cat) . n by A7, SEQ_1:12; then (abs Cat) . (n + 1) <= (4 * (abs r)) * ((abs Cat) . n) by A11, SEQ_1:12; then (abs Cat) . (n + 1) <= (4 * (abs r)) * (((4 * (abs r)) GeoSeq) . n) by A6, XXREAL_0:2; hence S2[n + 1] by A7, PREPOWER:3; ::_thesis: verum end; Cat . 0 = (Catalan (0 + 1)) * (r |^ 0) by A3; then A12: (abs Cat) . 0 = abs (r |^ 0) by CATALAN1:11, SEQ_1:12; ( r |^ 0 = 1 & (4 * (abs r)) |^ 0 = 1 ) by NEWTON:4; then A13: S2[ 0 ] by A12, A2, PREPOWER:def_1; for n being Nat holds S2[n] from NAT_1:sch_2(A13, A4); then A14: for n being Element of NAT holds S2[n] ; A15: now__::_thesis:_for_n_being_Element_of_NAT_holds_(abs_Cat)_._n_>=_0 let n be Element of NAT ; ::_thesis: (abs Cat) . n >= 0 (abs Cat) . n = abs (Cat . n) by SEQ_1:12; hence (abs Cat) . n >= 0 by COMPLEX1:46; ::_thesis: verum end; take Sum Cat ; ::_thesis: ( Sum Cat in REAL & S1[x, Sum Cat] ) thus Sum Cat in REAL ; ::_thesis: S1[x, Sum Cat] let s be real number ; ::_thesis: ( x = s implies ex Catal being Real_Sequence st ( ( for n being Nat holds Catal . n = (Catalan (n + 1)) * (s |^ n) ) & ( abs s < 1 / 4 implies ( Catal is absolutely_summable & Sum Catal = 1 + (s * ((Sum Catal) |^ 2)) & Sum Cat = Sum Catal ) ) ) ) assume A16: x = s ; ::_thesis: ex Catal being Real_Sequence st ( ( for n being Nat holds Catal . n = (Catalan (n + 1)) * (s |^ n) ) & ( abs s < 1 / 4 implies ( Catal is absolutely_summable & Sum Catal = 1 + (s * ((Sum Catal) |^ 2)) & Sum Cat = Sum Catal ) ) ) for y being set st y in NAT holds (Cat ^\ 1) . y = (Cat (##) (r (#) Cat)) . y proof let y be set ; ::_thesis: ( y in NAT implies (Cat ^\ 1) . y = (Cat (##) (r (#) Cat)) . y ) assume y in NAT ; ::_thesis: (Cat ^\ 1) . y = (Cat (##) (r (#) Cat)) . y then reconsider n = y as Element of NAT ; set n1 = n + 1; consider Fr1 being XFinSequence of such that A17: dom Fr1 = n + 1 and A18: for i being Nat st i in n + 1 holds Fr1 . i = (Cat . i) * ((r (#) Cat) . (n -' i)) and A19: Sum Fr1 = (Cat (##) (r (#) Cat)) . n by Def4; consider Catal being XFinSequence of such that A20: Sum Catal = Catalan ((n + 1) + 1) and A21: dom Catal = n + 1 and A22: for j being Nat st j < n + 1 holds Catal . j = (Catalan (j + 1)) * (Catalan ((n + 1) -' j)) by Th39; rng Catal c= REAL by XBOOLE_1:1; then reconsider CatalR = Catal as XFinSequence of by RELAT_1:def_19; defpred S3[ set , set ] means for k being Nat st k = $1 holds $2 = (r |^ (n + 1)) * (Catal . k); A23: for k being Nat st k in n + 1 holds ex x being Element of REAL st S3[k,x] proof let k be Nat; ::_thesis: ( k in n + 1 implies ex x being Element of REAL st S3[k,x] ) assume k in n + 1 ; ::_thesis: ex x being Element of REAL st S3[k,x] take (r |^ (n + 1)) * (Catal . k) ; ::_thesis: S3[k,(r |^ (n + 1)) * (Catal . k)] thus S3[k,(r |^ (n + 1)) * (Catal . k)] ; ::_thesis: verum end; consider Fr2 being XFinSequence of such that A24: dom Fr2 = n + 1 and A25: for k being Nat st k in n + 1 holds S3[k,Fr2 . k] from STIRL2_1:sch_5(A23); A26: now__::_thesis:_for_k_being_Nat_st_k_in_dom_Fr2_holds_ Fr1_._k_=_Fr2_._k let k be Nat; ::_thesis: ( k in dom Fr2 implies Fr1 . k = Fr2 . k ) assume A27: k in dom Fr2 ; ::_thesis: Fr1 . k = Fr2 . k A28: k in NAT by ORDINAL1:def_12; A29: k < n + 1 by A24, A27, NAT_1:44; then A30: (n + 1) -' k = (n + 1) - k by XREAL_1:233; A31: n = k + (n - k) ; k <= n by A29, NAT_1:13; then A32: n -' k = n - k by XREAL_1:233; then Fr1 . k = (Cat . k) * ((r (#) Cat) . (n - k)) by A18, A24, A27 .= ((Catalan (k + 1)) * (r |^ k)) * ((r (#) Cat) . (n - k)) by A3, A28 .= ((Catalan (k + 1)) * (r |^ k)) * (r * (Cat . (n - k))) by A32, SEQ_1:9 .= ((Catalan (k + 1)) * (r |^ k)) * (r * ((Catalan ((n -' k) + 1)) * (r |^ (n -' k)))) by A3, A32 .= (((Catalan (k + 1)) * (Catalan ((n + 1) -' k))) * r) * ((r |^ k) * (r |^ (n -' k))) by A32, A30 .= (((Catalan (k + 1)) * (Catalan ((n + 1) -' k))) * r) * (r |^ n) by A32, A31, NEWTON:8 .= ((Catal . k) * r) * (r |^ n) by A22, A29 .= (Catal . k) * (r * (r |^ n)) .= (Catal . k) * (r |^ (n + 1)) by NEWTON:6 .= Fr2 . k by A24, A25, A27 ; hence Fr1 . k = Fr2 . k ; ::_thesis: verum end; for k being Nat st k in len Fr2 holds Fr2 . k = (r |^ (n + 1)) * (CatalR . k) by A24, A25; then Sum Fr2 = (r |^ (n + 1)) * (Sum CatalR) by A21, A24, Th44 .= Cat . (n + 1) by A3, A20 .= (Cat ^\ 1) . n by NAT_1:def_3 ; hence (Cat ^\ 1) . y = (Cat (##) (r (#) Cat)) . y by A17, A19, A24, A26, AFINSQ_1:8; ::_thesis: verum end; then A33: Cat ^\ 1 = Cat (##) (r (#) Cat) by FUNCT_2:12; abs r >= 0 by COMPLEX1:46; then A34: abs (4 * (abs r)) = 4 * (abs r) by ABSVALUE:def_1; take Cat ; ::_thesis: ( ( for n being Nat holds Cat . n = (Catalan (n + 1)) * (s |^ n) ) & ( abs s < 1 / 4 implies ( Cat is absolutely_summable & Sum Cat = 1 + (s * ((Sum Cat) |^ 2)) & Sum Cat = Sum Cat ) ) ) hereby ::_thesis: ( abs s < 1 / 4 implies ( Cat is absolutely_summable & Sum Cat = 1 + (s * ((Sum Cat) |^ 2)) & Sum Cat = Sum Cat ) ) let n be Nat; ::_thesis: Cat . n = (Catalan (n + 1)) * (s |^ n) n in NAT by ORDINAL1:def_12; hence Cat . n = (Catalan (n + 1)) * (s |^ n) by A3, A16; ::_thesis: verum end; A35: r |^ 0 = 1 by NEWTON:4; Cat . 0 = (Catalan (0 + 1)) * (r |^ 0) by A3; then A36: (Partial_Sums Cat) . 0 = 1 by A35, CATALAN1:11, SERIES_1:def_1; assume abs s < 1 / 4 ; ::_thesis: ( Cat is absolutely_summable & Sum Cat = 1 + (s * ((Sum Cat) |^ 2)) & Sum Cat = Sum Cat ) then 4 * (abs r) < 4 * (1 / 4) by A16, XREAL_1:68; then (4 * (abs r)) GeoSeq is summable by A34, SERIES_1:24; then abs Cat is summable by A15, A14, SERIES_1:20; hence A37: Cat is absolutely_summable by SERIES_1:def_4; ::_thesis: ( Sum Cat = 1 + (s * ((Sum Cat) |^ 2)) & Sum Cat = Sum Cat ) then Cat is summable ; then ( r (#) Cat is summable & Sum (r (#) Cat) = r * (Sum Cat) ) by SERIES_1:10; then Sum (Cat ^\ (0 + 1)) = (Sum Cat) * (r * (Sum Cat)) by A37, A33, Th53; then Sum Cat = 1 + (r * ((Sum Cat) * (Sum Cat))) by A37, A36, SERIES_1:15; hence ( Sum Cat = 1 + (s * ((Sum Cat) |^ 2)) & Sum Cat = Sum Cat ) by A16, WSIERP_1:1; ::_thesis: verum end; consider SumC being Function of REAL,REAL such that A38: for x being set st x in REAL holds S1[x,SumC . x] from FUNCT_2:sch_1(A1); A39: for r, s being real number st 0 < s & s <= r & r < 1 / 4 holds SumC . s <= SumC . r proof let r, s be real number ; ::_thesis: ( 0 < s & s <= r & r < 1 / 4 implies SumC . s <= SumC . r ) assume that A40: 0 < s and A41: s <= r and A42: r < 1 / 4 ; ::_thesis: SumC . s <= SumC . r r is Real by XREAL_0:def_1; then consider Cr being Real_Sequence such that A43: for n being Nat holds Cr . n = (Catalan (n + 1)) * (r |^ n) and A44: ( abs r < 1 / 4 implies ( Cr is absolutely_summable & Sum Cr = 1 + (r * ((Sum Cr) |^ 2)) & SumC . r = Sum Cr ) ) by A38; s is Real by XREAL_0:def_1; then consider Cs being Real_Sequence such that A45: for n being Nat holds Cs . n = (Catalan (n + 1)) * (s |^ n) and A46: ( abs s < 1 / 4 implies ( Cs is absolutely_summable & Sum Cs = 1 + (s * ((Sum Cs) |^ 2)) & SumC . s = Sum Cs ) ) by A38; A47: now__::_thesis:_for_n_being_Element_of_NAT_holds_Cs_._n_<=_Cr_._n let n be Element of NAT ; ::_thesis: Cs . n <= Cr . n ( s |^ n <= r |^ n & Catalan (n + 1) >= 0 ) by A40, A41, CATALAN1:17, PREPOWER:9; then A48: (Catalan (n + 1)) * (s |^ n) <= (Catalan (n + 1)) * (r |^ n) by XREAL_1:64; (Catalan (n + 1)) * (r |^ n) = Cr . n by A43; hence Cs . n <= Cr . n by A45, A48; ::_thesis: verum end; A49: s < 1 / 4 by A41, A42, XXREAL_0:2; thus SumC . s <= SumC . r by A40, A41, A42, A49, A46, A44, A47, ABSVALUE:def_1, TIETZE:5; ::_thesis: verum end; set R = { r where r is Real : ( 0 < r & r < 1 / 4 & SumC . r = (1 + (sqrt (1 - (4 * r)))) / (2 * r) ) } ; A50: for r being real number st r <> 0 & abs r < 1 / 4 & not SumC . r = (1 - (sqrt (1 - (4 * r)))) / (2 * r) holds SumC . r = (1 + (sqrt (1 - (4 * r)))) / (2 * r) proof let r be real number ; ::_thesis: ( r <> 0 & abs r < 1 / 4 & not SumC . r = (1 - (sqrt (1 - (4 * r)))) / (2 * r) implies SumC . r = (1 + (sqrt (1 - (4 * r)))) / (2 * r) ) assume that A51: r <> 0 and A52: abs r < 1 / 4 ; ::_thesis: ( SumC . r = (1 - (sqrt (1 - (4 * r)))) / (2 * r) or SumC . r = (1 + (sqrt (1 - (4 * r)))) / (2 * r) ) r <= 1 / 4 by A52, ABSVALUE:5; then 4 * r <= (1 / 4) * 4 by XREAL_1:64; then A53: (4 * r) - (4 * r) <= 1 - (4 * r) by XREAL_1:9; r is Real by XREAL_0:def_1; then consider Catal being Real_Sequence such that for n being Nat holds Catal . n = (Catalan (n + 1)) * (r |^ n) and A54: ( abs r < 1 / 4 implies ( Catal is absolutely_summable & Sum Catal = 1 + (r * ((Sum Catal) |^ 2)) & SumC . r = Sum Catal ) ) by A38; set S = Sum Catal; Sum Catal = 1 + (r * ((Sum Catal) ^2)) by A52, A54, WSIERP_1:1; then A55: ((r * ((Sum Catal) ^2)) + ((- 1) * (Sum Catal))) + 1 = 0 ; ( delta (r,(- 1),1) = 1 - (4 * r) & - (- 1) = 1 ) ; hence ( SumC . r = (1 - (sqrt (1 - (4 * r)))) / (2 * r) or SumC . r = (1 + (sqrt (1 - (4 * r)))) / (2 * r) ) by A51, A52, A54, A55, A53, FIB_NUM:6; ::_thesis: verum end; A56: for r, s being real number st 0 < r & r < s & s < 1 / 4 holds (1 + (sqrt (1 - (4 * r)))) / (2 * r) > (1 + (sqrt (1 - (4 * s)))) / (2 * s) proof let r, s be real number ; ::_thesis: ( 0 < r & r < s & s < 1 / 4 implies (1 + (sqrt (1 - (4 * r)))) / (2 * r) > (1 + (sqrt (1 - (4 * s)))) / (2 * s) ) assume that A57: 0 < r and A58: r < s and A59: s < 1 / 4 ; ::_thesis: (1 + (sqrt (1 - (4 * r)))) / (2 * r) > (1 + (sqrt (1 - (4 * s)))) / (2 * s) 4 * s < 4 * (1 / 4) by A59, XREAL_1:68; then A60: (4 * s) - (4 * s) < 1 - (4 * s) by XREAL_1:9; then A61: sqrt (1 - (4 * s)) > 0 by SQUARE_1:25; 4 * r < 4 * s by A58, XREAL_1:68; then 1 - (4 * r) >= 1 - (4 * s) by XREAL_1:10; then sqrt (1 - (4 * r)) >= sqrt (1 - (4 * s)) by A60, SQUARE_1:26; then 1 + (sqrt (1 - (4 * r))) >= 1 + (sqrt (1 - (4 * s))) by XREAL_1:7; then A62: (1 + (sqrt (1 - (4 * r)))) / (2 * r) >= (1 + (sqrt (1 - (4 * s)))) / (2 * r) by A57, XREAL_1:72; ( 2 * r > 2 * 0 & 2 * r < 2 * s ) by A57, A58, XREAL_1:68; then (1 + (sqrt (1 - (4 * s)))) / (2 * r) > (1 + (sqrt (1 - (4 * s)))) / (2 * s) by A61, XREAL_1:76; hence (1 + (sqrt (1 - (4 * r)))) / (2 * r) > (1 + (sqrt (1 - (4 * s)))) / (2 * s) by A62, XXREAL_0:2; ::_thesis: verum end; A63: { r where r is Real : ( 0 < r & r < 1 / 4 & SumC . r = (1 + (sqrt (1 - (4 * r)))) / (2 * r) ) } = {} proof assume { r where r is Real : ( 0 < r & r < 1 / 4 & SumC . r = (1 + (sqrt (1 - (4 * r)))) / (2 * r) ) } <> {} ; ::_thesis: contradiction then consider x being set such that A64: x in { r where r is Real : ( 0 < r & r < 1 / 4 & SumC . r = (1 + (sqrt (1 - (4 * r)))) / (2 * r) ) } by XBOOLE_0:def_1; consider r being Real such that x = r and A65: 0 < r and A66: r < 1 / 4 and A67: SumC . r = (1 + (sqrt (1 - (4 * r)))) / (2 * r) by A64; consider s being real number such that A68: r < s and A69: s < 1 / 4 by A66, XREAL_1:5; A70: abs s = s by A65, A68, ABSVALUE:def_1; 4 * s < 4 * (1 / 4) by A69, XREAL_1:68; then (4 * s) - (4 * s) < 1 - (4 * s) by XREAL_1:9; then sqrt (1 - (4 * s)) > 0 by SQUARE_1:25; then 1 - (sqrt (1 - (4 * s))) <= 1 - 0 by XREAL_1:10; then A71: (1 - (sqrt (1 - (4 * s)))) / (2 * s) <= 1 / (2 * s) by A65, A68, XREAL_1:72; A72: 2 * r > 2 * 0 by A65, XREAL_1:68; A73: s is Real by XREAL_0:def_1; { r where r is Real : ( 0 < r & r < 1 / 4 & SumC . r = (1 + (sqrt (1 - (4 * r)))) / (2 * r) ) } c= {r} proof assume not { r where r is Real : ( 0 < r & r < 1 / 4 & SumC . r = (1 + (sqrt (1 - (4 * r)))) / (2 * r) ) } c= {r} ; ::_thesis: contradiction then { r where r is Real : ( 0 < r & r < 1 / 4 & SumC . r = (1 + (sqrt (1 - (4 * r)))) / (2 * r) ) } \ {r} <> {} by XBOOLE_1:37; then consider y being set such that A74: y in { r where r is Real : ( 0 < r & r < 1 / 4 & SumC . r = (1 + (sqrt (1 - (4 * r)))) / (2 * r) ) } \ {r} by XBOOLE_0:def_1; y in { r where r is Real : ( 0 < r & r < 1 / 4 & SumC . r = (1 + (sqrt (1 - (4 * r)))) / (2 * r) ) } by A74; then consider s being Real such that A75: y = s and A76: 0 < s and A77: s < 1 / 4 and A78: SumC . s = (1 + (sqrt (1 - (4 * s)))) / (2 * s) ; A79: r <> s by A74, A75, ZFMISC_1:56; now__::_thesis:_contradiction percases ( r > s or r < s ) by A79, XXREAL_0:1; supposeA80: r > s ; ::_thesis: contradiction then SumC . s > SumC . r by A56, A66, A67, A76, A78; hence contradiction by A39, A66, A76, A80; ::_thesis: verum end; supposeA81: r < s ; ::_thesis: contradiction then SumC . r > SumC . s by A56, A65, A67, A77, A78; hence contradiction by A39, A65, A77, A81; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; then not s in { r where r is Real : ( 0 < r & r < 1 / 4 & SumC . r = (1 + (sqrt (1 - (4 * r)))) / (2 * r) ) } by A68, TARSKI:def_1; then SumC . s <> (1 + (sqrt (1 - (4 * s)))) / (2 * s) by A65, A68, A69, A73; then A82: SumC . s = (1 - (sqrt (1 - (4 * s)))) / (2 * s) by A50, A65, A68, A69, A70; 4 * r < 4 * (1 / 4) by A66, XREAL_1:68; then (4 * r) - (4 * r) < 1 - (4 * r) by XREAL_1:9; then sqrt (1 - (4 * r)) > 0 by SQUARE_1:25; then 1 + 0 < 1 + (sqrt (1 - (4 * r))) by XREAL_1:8; then A83: 1 / (2 * r) < SumC . r by A67, A72, XREAL_1:74; 2 * r < 2 * s by A68, XREAL_1:68; then 1 / (2 * s) < 1 / (2 * r) by A72, XREAL_1:76; then SumC . s < 1 / (2 * r) by A82, A71, XXREAL_0:2; then SumC . s < SumC . r by A83, XXREAL_0:2; hence contradiction by A39, A65, A68, A69; ::_thesis: verum end; A84: for r being real number st 0 < r & abs r < 1 / 4 holds SumC . r = (1 - (sqrt (1 - (4 * r)))) / (2 * r) proof let r be real number ; ::_thesis: ( 0 < r & abs r < 1 / 4 implies SumC . r = (1 - (sqrt (1 - (4 * r)))) / (2 * r) ) assume that A85: 0 < r and A86: abs r < 1 / 4 ; ::_thesis: SumC . r = (1 - (sqrt (1 - (4 * r)))) / (2 * r) assume SumC . r <> (1 - (sqrt (1 - (4 * r)))) / (2 * r) ; ::_thesis: contradiction then A87: SumC . r = (1 + (sqrt (1 - (4 * r)))) / (2 * r) by A50, A85, A86; abs r = r by A85, ABSVALUE:def_1; then r in { r where r is Real : ( 0 < r & r < 1 / 4 & SumC . r = (1 + (sqrt (1 - (4 * r)))) / (2 * r) ) } by A85, A86, A87; hence contradiction by A63; ::_thesis: verum end; A88: for r being real number st r < 0 & abs r < 1 / 4 holds SumC . r = (1 - (sqrt (1 - (4 * r)))) / (2 * r) proof let r be real number ; ::_thesis: ( r < 0 & abs r < 1 / 4 implies SumC . r = (1 - (sqrt (1 - (4 * r)))) / (2 * r) ) assume that A89: r < 0 and A90: abs r < 1 / 4 ; ::_thesis: SumC . r = (1 - (sqrt (1 - (4 * r)))) / (2 * r) 2 * r < 2 * 0 by A89, XREAL_1:68; then A91: ( abs (2 * r) = - (2 * r) & 0 - (2 * r) > 0 - 0 ) by ABSVALUE:def_1; A92: abs (- r) < 1 / 4 by A90, COMPLEX1:52; then 1 / 4 >= - r by ABSVALUE:5; then 4 * (1 / 4) >= 4 * (- r) by XREAL_1:64; then 1 - (4 * (- r)) >= (4 * (- r)) - (4 * (- r)) by XREAL_1:9; then sqrt (1 - (4 * (- r))) >= 0 by SQUARE_1:def_2; then A93: 1 - (sqrt (1 - (4 * (- r)))) <= 1 - 0 by XREAL_1:10; A94: sqrt (1 - (4 * r)) > 0 by A89, SQUARE_1:25; then 1 + (sqrt (1 - (4 * r))) > 1 + 0 by XREAL_1:8; then A95: 1 - (sqrt (1 - (4 * (- r)))) < 1 + (sqrt (1 - (4 * r))) by A93, XXREAL_0:2; 1 + (sqrt (1 - (4 * r))) = abs (1 + (sqrt (1 - (4 * r)))) by A94, ABSVALUE:def_1; then A96: (1 - (sqrt (1 - (4 * (- r))))) / (2 * (- r)) < (abs (1 + (sqrt (1 - (4 * r))))) / (abs (2 * r)) by A95, A91, XREAL_1:74; consider CR being Real_Sequence such that A97: for n being Nat holds CR . n = (Catalan (n + 1)) * ((- r) |^ n) and A98: ( abs (- r) < 1 / 4 implies ( CR is absolutely_summable & Sum CR = 1 + ((- r) * ((Sum CR) |^ 2)) & SumC . (- r) = Sum CR ) ) by A38; assume A99: SumC . r <> (1 - (sqrt (1 - (4 * r)))) / (2 * r) ; ::_thesis: contradiction r is Real by XREAL_0:def_1; then consider Cr being Real_Sequence such that A100: for n being Nat holds Cr . n = (Catalan (n + 1)) * (r |^ n) and A101: ( abs r < 1 / 4 implies ( Cr is absolutely_summable & Sum Cr = 1 + (r * ((Sum Cr) |^ 2)) & SumC . r = Sum Cr ) ) by A38; now__::_thesis:_for_x_being_set_st_x_in_NAT_holds_ (abs_Cr)_._x_=_CR_._x let x be set ; ::_thesis: ( x in NAT implies (abs Cr) . x = CR . x ) assume x in NAT ; ::_thesis: (abs Cr) . x = CR . x then reconsider n = x as Element of NAT ; (- r) |^ n = ((- 1) * r) |^ n .= ((- 1) |^ n) * (r |^ n) by NEWTON:7 ; then A102: abs ((- r) |^ n) = (abs ((- 1) |^ n)) * (abs (r |^ n)) by COMPLEX1:65 .= 1 * (abs (r |^ n)) by SERIES_2:1 ; Catalan (n + 1) >= 0 by CATALAN1:17; then A103: abs (Catalan (n + 1)) = Catalan (n + 1) by ABSVALUE:def_1; (- r) |^ n >= 0 by A89, POWER:3; then abs ((- r) |^ n) = (- r) |^ n by ABSVALUE:def_1; then CR . n = (abs (r |^ n)) * (abs (Catalan (n + 1))) by A97, A102, A103 .= abs ((r |^ n) * (Catalan (n + 1))) by COMPLEX1:65 .= abs (Cr . n) by A100 .= (abs Cr) . n by SEQ_1:12 ; hence (abs Cr) . x = CR . x ; ::_thesis: verum end; then A104: abs Cr = CR by FUNCT_2:12; 0 - r > 0 - 0 by A89; then A105: Sum CR = (1 - (sqrt (1 - (4 * (- r))))) / (2 * (- r)) by A84, A98, A92; abs (Sum Cr) <= Sum (abs Cr) by A90, A101, TIETZE:6; then abs ((1 + (sqrt (1 - (4 * r)))) / (2 * r)) <= Sum CR by A50, A89, A90, A101, A104, A99; hence contradiction by A105, A96, COMPLEX1:67; ::_thesis: verum end; let r be real number ; ::_thesis: ex Catal being Real_Sequence st ( ( for n being Nat holds Catal . n = (Catalan (n + 1)) * (r |^ n) ) & ( abs r < 1 / 4 implies ( Catal is absolutely_summable & Sum Catal = 1 + (r * ((Sum Catal) |^ 2)) & Sum Catal = 2 / (1 + (sqrt (1 - (4 * r)))) & ( r <> 0 implies Sum Catal = (1 - (sqrt (1 - (4 * r)))) / (2 * r) ) ) ) ) r is Real by XREAL_0:def_1; then consider Cat being Real_Sequence such that A106: for n being Nat holds Cat . n = (Catalan (n + 1)) * (r |^ n) and A107: ( abs r < 1 / 4 implies ( Cat is absolutely_summable & Sum Cat = 1 + (r * ((Sum Cat) |^ 2)) & SumC . r = Sum Cat ) ) by A38; set s = sqrt (1 - (4 * r)); take Cat ; ::_thesis: ( ( for n being Nat holds Cat . n = (Catalan (n + 1)) * (r |^ n) ) & ( abs r < 1 / 4 implies ( Cat is absolutely_summable & Sum Cat = 1 + (r * ((Sum Cat) |^ 2)) & Sum Cat = 2 / (1 + (sqrt (1 - (4 * r)))) & ( r <> 0 implies Sum Cat = (1 - (sqrt (1 - (4 * r)))) / (2 * r) ) ) ) ) thus for n being Nat holds Cat . n = (Catalan (n + 1)) * (r |^ n) by A106; ::_thesis: ( abs r < 1 / 4 implies ( Cat is absolutely_summable & Sum Cat = 1 + (r * ((Sum Cat) |^ 2)) & Sum Cat = 2 / (1 + (sqrt (1 - (4 * r)))) & ( r <> 0 implies Sum Cat = (1 - (sqrt (1 - (4 * r)))) / (2 * r) ) ) ) assume A108: abs r < 1 / 4 ; ::_thesis: ( Cat is absolutely_summable & Sum Cat = 1 + (r * ((Sum Cat) |^ 2)) & Sum Cat = 2 / (1 + (sqrt (1 - (4 * r)))) & ( r <> 0 implies Sum Cat = (1 - (sqrt (1 - (4 * r)))) / (2 * r) ) ) hence ( Cat is absolutely_summable & Sum Cat = 1 + (r * ((Sum Cat) |^ 2)) ) by A107; ::_thesis: ( Sum Cat = 2 / (1 + (sqrt (1 - (4 * r)))) & ( r <> 0 implies Sum Cat = (1 - (sqrt (1 - (4 * r)))) / (2 * r) ) ) A109: ( r <> 0 implies Sum Cat = (1 - (sqrt (1 - (4 * r)))) / (2 * r) ) proof assume r <> 0 ; ::_thesis: Sum Cat = (1 - (sqrt (1 - (4 * r)))) / (2 * r) then ( r > 0 or r < 0 ) ; hence Sum Cat = (1 - (sqrt (1 - (4 * r)))) / (2 * r) by A84, A88, A107, A108; ::_thesis: verum end; now__::_thesis:_2_/_(1_+_(sqrt_(1_-_(4_*_r))))_=_Sum_Cat percases ( r = 0 or r <> 0 ) ; suppose r = 0 ; ::_thesis: 2 / (1 + (sqrt (1 - (4 * r)))) = Sum Cat hence 2 / (1 + (sqrt (1 - (4 * r)))) = Sum Cat by A107, A108, SQUARE_1:18; ::_thesis: verum end; supposeA110: r <> 0 ; ::_thesis: Sum Cat = 2 / (1 + (sqrt (1 - (4 * r)))) then A111: 2 * r <> 0 ; r <= 1 / 4 by A108, ABSVALUE:5; then 4 * r <= 4 * (1 / 4) by XREAL_1:64; then A112: 1 - (4 * r) >= (4 * r) - (4 * r) by XREAL_1:9; then sqrt (1 - (4 * r)) >= 0 by SQUARE_1:def_2; then (1 + (sqrt (1 - (4 * r)))) / (1 + (sqrt (1 - (4 * r)))) = 1 by XCMPLX_1:60; then (1 - (sqrt (1 - (4 * r)))) / (2 * r) = ((1 - (sqrt (1 - (4 * r)))) / (2 * r)) * ((1 + (sqrt (1 - (4 * r)))) / (1 + (sqrt (1 - (4 * r))))) .= ((1 - (sqrt (1 - (4 * r)))) * (1 + (sqrt (1 - (4 * r))))) / ((2 * r) * (1 + (sqrt (1 - (4 * r))))) by XCMPLX_1:76 .= ((1 ^2) - ((sqrt (1 - (4 * r))) ^2)) / ((2 * r) * (1 + (sqrt (1 - (4 * r))))) .= (1 - (1 - (4 * r))) / ((2 * r) * (1 + (sqrt (1 - (4 * r))))) by A112, SQUARE_1:def_2 .= ((2 * r) * 2) / ((2 * r) * (1 + (sqrt (1 - (4 * r))))) .= ((2 * r) / (2 * r)) * (2 / (1 + (sqrt (1 - (4 * r))))) by XCMPLX_1:76 .= 1 * (2 / (1 + (sqrt (1 - (4 * r))))) by A111, XCMPLX_1:60 ; hence Sum Cat = 2 / (1 + (sqrt (1 - (4 * r)))) by A109, A110; ::_thesis: verum end; end; end; hence ( Sum Cat = 2 / (1 + (sqrt (1 - (4 * r)))) & ( r <> 0 implies Sum Cat = (1 - (sqrt (1 - (4 * r)))) / (2 * r) ) ) by A109; ::_thesis: verum end;